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A080851
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Square array of pyramidal numbers, read by antidiagonals.
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28
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1, 1, 3, 1, 4, 6, 1, 5, 10, 10, 1, 6, 14, 20, 15, 1, 7, 18, 30, 35, 21, 1, 8, 22, 40, 55, 56, 28, 1, 9, 26, 50, 75, 91, 84, 36, 1, 10, 30, 60, 95, 126, 140, 120, 45, 1, 11, 34, 70, 115, 161, 196, 204, 165, 55, 1, 12, 38, 80, 135, 196, 252, 288, 285, 220, 66, 1, 13, 42, 90, 155, 231, 308, 372, 405, 385, 286, 78
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OFFSET
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0,3
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COMMENTS
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The first row contains the triangular numbers, which are really two-dimensional, but can be regarded as degenerate pyramidal numbers. - N. J. A. Sloane, Aug 28 2015
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LINKS
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FORMULA
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T(n, k) = binomial(k+3, 3) + (n-1)*binomial(k+2, 3), corrected Oct 01 2021.
T(n, k) = T(n-1, k) + C(k+2, 3) = T(n-1, k) + k*(k+1)*(k+2)/6.
G.f. for rows: (1 + n*x)/(1-x)^4, n>=-1.
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EXAMPLE
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Array begins (n>=0, k>=0):
1, 3, 6, 10, 15, 21, 28, 36, 45, 55, ... A000217
1, 4, 10, 20, 35, 56, 84, 120, 165, 220, ... A000292
1, 5, 14, 30, 55, 91, 140, 204, 285, 385, ... A000330
1, 6, 18, 40, 75, 126, 196, 288, 405, 550, ... A002411
1, 7, 22, 50, 95, 161, 252, 372, 525, 715, ... A002412
1, 8, 26, 60, 115, 196, 308, 456, 645, 880, ... A002413
1, 9, 30, 70, 135, 231, 364, 540, 765, 1045, ... A002414
1, 10, 34, 80, 155, 266, 420, 624, 885, 1210, ... A007584
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MAPLE
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binomial(k+3, 3)+(n-1)*binomial(k+2, 3) ;
end proc:
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MATHEMATICA
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pyramidalFigurative[ ngon_, rank_] := (3 rank^2 + rank^3 (ngon - 2) - rank (ngon - 5))/6; Table[ pyramidalFigurative[n-k-1, k], {n, 4, 15}, {k, n-3}] // Flatten (* Robert G. Wilson v, Sep 15 2015 *)
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PROG
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(Derive) vector(vector(poly_coeff(Taylor((1+kx)/(1-x)^4, x, 11), x, n), n, 0, 11), k, -1, 10) VECTOR(VECTOR(comb(k+2, 2)+comb(k+2, 3)n, k, 0, 11), n, 0, 11)
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CROSSREFS
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See A257199 for another version of this array.
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KEYWORD
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AUTHOR
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STATUS
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approved
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