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A176244
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Triangle generated by T(n,k) = q^k*T(n-1, k) + T(n-1, k-1), with q=4.
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3
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1, 1, 1, 1, 17, 1, 1, 273, 81, 1, 1, 4369, 5457, 337, 1, 1, 69905, 353617, 91729, 1361, 1, 1, 1118481, 22701393, 23836241, 1485393, 5457, 1, 1, 17895697, 1454007633, 6124779089, 1544878673, 23837265, 21841, 1, 1, 286331153, 93074384209, 1569397454417, 1588080540241, 99182316113, 381680209, 87377, 1
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OFFSET
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1,5
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COMMENTS
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Row sums are: {1, 2, 19, 356, 10165, 516614, 49146967, 9165420200, 3350402793721, 2449781908163402, ...}.
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REFERENCES
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Steve Roman, The Umbral Calculus, Dover Publications, New York (1984), page 176
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LINKS
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FORMULA
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T(n,k) = T(n-1, k-1) + q^k*T(n-1, k), with q=4.
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EXAMPLE
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Triangle starts as:
1;
1, 1;
1, 17, 1;
1, 273, 81, 1;
1, 4369, 5457, 337, 1;
1, 69905, 353617, 91729, 1361, 1;
1, 1118481, 22701393, 23836241, 1485393, 5457, 1;
1, 17895697, 1454007633, 6124779089, 1544878673, 23837265, 21841, 1;
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MAPLE
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T:= proc(n, k) option remember;
q:=4;
if k=1 or k=n then 1
else T(n-1, k-1) + q^k*T(n-1, k)
fi; end:
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MATHEMATICA
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q:=4; T[n_, k_]:= T[n, k]= If[k==1 || k==n, 1, q^k*T[n-1, k] + T[n-1, k-1]];
Table[T[n, k], {n, 12}, {k, n}]//Flatten (* G. C. Greubel, Nov 22 2019 *)
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PROG
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(PARI) T(n, k) = my(q=4); if(k==1 || k==n, 1, q^k*T(n-1, k) + T(n-1, k-1)); \\ G. C. Greubel, Nov 22 2019
(Magma)
function T(n, k)
q:=4;
if k eq 1 or k eq n then return 1;
else return T(n-1, k-1) + q^k*T(n-1, k);
end if; return T; end function;
[T(n, k): k in [1..n], n in [1..12]]; // G. C. Greubel, Nov 22 2019
(Sage)
@CachedFunction
def T(n, k):
q=4;
if (k==1 or k==n): return 1
else: return q^k*T(n-1, k) + T(n-1, k-1)
[[T(n, k) for k in (1..n)] for n in (1..12)] # G. C. Greubel, Nov 22 2019
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CROSSREFS
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KEYWORD
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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