Revision History for A209695
(Underlined text is an addition;
strikethrough text is a deletion.)
Showing entries 1-10
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#16 by Giovanni Resta at Fri Jan 24 03:28:36 EST 2020
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#15 by Michel Marcus at Fri Jan 24 00:26:58 EST 2020
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#14 by Jon E. Schoenfield at Thu Jan 23 21:55:41 EST 2020
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#13 by Jon E. Schoenfield at Thu Jan 23 21:55:39 EST 2020
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| FORMULA
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u(n,x)=) = x*u(n-1,x)+() + (x+1)*v(n-1,x),
v(n,x)=) = 2x*u(n-1,x)+() + (x+1)*v(n-1,x),
Contribution fromFrom Philippe Deléham, Mar 24 2012. (: (Start)
As DELTA-triangle T(n,k) with 0<= <= k<= <= n:
T(n,k) = T(n-1,k) + 2*T(n-1,k-1) + T(n-2,k-1) + T(n-2,k-2), T(0,0) = T(1,0) = T(2,0) = 1, T(1,1) = T(2,2) = 0, T(2,1) = 2 and T(n,k) = 0 if k< < 0 or if k> > n. (End)
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| EXAMPLE
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1;
1..., 2;
1..., 5...., 5;
1..., 8...., 18..., 12;
1..., 11..., 40..., 58..., 29;
First three polynomials u(n,x): 1, 1 + 2x, 1 + 5x + 5x^2.):
1
1 + 2x
1 + 5x + 5x^2.
From Philippe Deléham, Mar 24 2012: (Start)
(1, 0, 1/2, -1/2, 0, 0, ...) DELTA (0, 2, 1/2, -1/2, 0, 0, ...) begins ::
1;
1, , 0;
1, , 2, , 0;
1, , 5, , 5, , 0;
1, , 8, 18, 12, , 0;
1, 11, 40, 58, 29, 0 . - _Philippe Deléham_, Mar 24 2012; (End)
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approved
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#12 by N. J. A. Sloane at Sun Sep 08 19:59:32 EDT 2013
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| COMMENTS
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Subtriangle of the triangle given by (1, 0, 1/2, -1/2, 0, 0, 0, 0, 0, 0, ...) DELTA (0, 2, 1/2, -1/2, 0, 0, 0, 0, 0, 0, 0, ...) where DELTA is the operator defined in A084938. - . - _Philippe Deléham, _, Mar 24 2012
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| FORMULA
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Contribution from _Philippe Deléham, _, Mar 24 2012. (Start)
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| EXAMPLE
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1, 11, 40, 58, 29, 0 . - . - _Philippe Deléham, _, Mar 24 2012
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Discussion
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| Sun Sep 08
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| OEIS Server: https://oeis.org/edit/global/1941
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#11 by N. J. A. Sloane at Fri Feb 22 14:40:35 EST 2013
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| COMMENTS
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Subtriangle of the triangle given by (1, 0, 1/2, -1/2, 0, 0, 0, 0, 0, 0, ...) DELTA (0, 2, 1/2, -1/2, 0, 0, 0, 0, 0, 0, 0, ...) where DELTA is the operator defined in A084938. - DELEHAM Philippe Deléham, Mar 24 2012
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| FORMULA
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Contribution from DELEHAM Philippe Deléham, Mar 24 2012. (Start)
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| EXAMPLE
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1, 11, 40, 58, 29, 0 . - DELEHAM Philippe Deléham, Mar 24 2012
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Discussion
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| Fri Feb 22
| 14:40
| OEIS Server: https://oeis.org/edit/global/1863
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#10 by Russ Cox at Fri Mar 30 18:58:15 EDT 2012
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| AUTHOR
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_Clark Kimberling (ck6(AT)evansville.edu), _, Mar 13 2012
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Discussion
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| Fri Mar 30
| 18:58
| OEIS Server: https://oeis.org/edit/global/285
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#9 by T. D. Noe at Sat Mar 24 12:54:41 EDT 2012
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#8 by DELEHAM Philippe at Sat Mar 24 03:22:36 EDT 2012
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#7 by DELEHAM Philippe at Sat Mar 24 03:22:29 EDT 2012
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| COMMENTS
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Subtriangle of the triangle given by (1, 0, 1/2, -1/2, 0, 0, 0, 0, 0, 0, ...) DELTA (0, 2, 1/2, -1/2, 0, 0, 0, 0, 0, 0, 0, ...) where DELTA is the operator defined in A084938. - DELEHAM Philippe, Mar 24 2012
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| FORMULA
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Contribution from DELEHAM Philippe, Mar 24 2012. (Start)
As DELTA-triangle T(n,k) with 0<=k<=n:
G.f.: (1-2*y*x-y*x^2-y^2*x^2)/(1-x-2*y*x-y*x^2-y^2*x^2).
T(n,k) = T(n-1,k) + 2*T(n-1,k-1) + T(n-2,k-1) + T(n-2,k-2), T(0,0) = T(1,0) = T(2,0) = 1, T(1,1) = T(2,2) = 0, T(2,1) = 2 and T(n,k) = 0 if k<0 or if k>n. (End)
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| EXAMPLE
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(1, 0, 1/2, -1/2, 0, 0, ...) DELTA (0, 2, 1/2, -1/2, 0, 0, ...) begins :
1
1, 0
1, 2, 0
1, 5, 5, 0
1, 8, 18, 12, 0
1, 11, 40, 58, 29, 0 . - DELEHAM Philippe, Mar 24 2012
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approved
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