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A080851
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Square array of pyramidal numbers, read by antidiagonals.
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18
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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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Alois P. Heinz, Antidiagonals n = 0..140, flattened
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FORMULA
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T(n, k) = C(k+2, 2) + n*C(k+2, 3).
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.
T(n,k) = sum_{j=1..k+1} A057145(n+2,j). - R. J. Mathar, Jul 28 2016
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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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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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Numerous sequences in the database are to be found in the array. Rows include the pyramidal numbers A000217, A000292, A000330, A002411, A002412, A002413, A002414, A007584, A007585, A007586.
Columns include or are closely related to A017029, A017113, A017017, A017101, A016777, A017305. Diagonals include A006325, A006484, A002417.
Cf. A057145.
See A257199 for another version of this array.
Sequence in context: A194540 A193043 A086271 * A209518 A108285 A207619
Adjacent sequences: A080848 A080849 A080850 * A080852 A080853 A080854
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KEYWORD
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nonn,tabl,easy
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AUTHOR
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Paul Barry, Feb 21 2003
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STATUS
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approved
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