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%I #35 Jan 19 2024 20:16:34
%S 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,
%T 28,1,9,26,50,75,91,84,36,1,10,30,60,95,126,140,120,45,1,11,34,70,115,
%U 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
%N Square array of pyramidal numbers, read by antidiagonals.
%C 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
%H Alois P. Heinz, <a href="/A080851/b080851.txt">Antidiagonals n = 0..140, flattened</a>
%F T(n, k) = binomial(k+3, 3) + (n-1)*binomial(k+2, 3), corrected Oct 01 2021.
%F T(n, k) = T(n-1, k) + C(k+2, 3) = T(n-1, k) + k*(k+1)*(k+2)/6.
%F G.f. for rows: (1 + n*x)/(1-x)^4, n>=-1.
%F T(n,k) = sum_{j=1..k+1} A057145(n+2,j). - _R. J. Mathar_, Jul 28 2016
%e Array begins (n>=0, k>=0):
%e 1, 3, 6, 10, 15, 21, 28, 36, 45, 55, ... A000217
%e 1, 4, 10, 20, 35, 56, 84, 120, 165, 220, ... A000292
%e 1, 5, 14, 30, 55, 91, 140, 204, 285, 385, ... A000330
%e 1, 6, 18, 40, 75, 126, 196, 288, 405, 550, ... A002411
%e 1, 7, 22, 50, 95, 161, 252, 372, 525, 715, ... A002412
%e 1, 8, 26, 60, 115, 196, 308, 456, 645, 880, ... A002413
%e 1, 9, 30, 70, 135, 231, 364, 540, 765, 1045, ... A002414
%e 1, 10, 34, 80, 155, 266, 420, 624, 885, 1210, ... A007584
%p A080851 := proc(n,k)
%p binomial(k+3,3)+(n-1)*binomial(k+2,3) ;
%p end proc:
%p seq( seq(A080851(d-k,k),k=0..d),d=0..12) ; # _R. J. Mathar_, Oct 01 2021
%t 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 *)
%o (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)
%Y 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.
%Y Columns include or are closely related to A017029, A017113, A017017, A017101, A016777, A017305. Diagonals include A006325, A006484, A002417.
%Y Cf. A057145, A027660 (antidiagonal sums).
%Y See A257199 for another version of this array.
%K nonn,tabl,easy
%O 0,3
%A _Paul Barry_, Feb 21 2003
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