A317023 Square array A(n,k), n >= 0, k >= 0, read by ascending antidiagonals, where the sequence of row n is the expansion of (1-x^(n+1))/((1-x)^(n+1)).
1, 1, 0, 1, 2, 0, 1, 3, 2, 0, 1, 4, 6, 2, 0, 1, 5, 10, 9, 2, 0, 1, 6, 15, 20, 12, 2, 0, 1, 7, 21, 35, 34, 15, 2, 0, 1, 8, 28, 56, 70, 52, 18, 2, 0, 1, 9, 36, 84, 126, 125, 74, 21, 2, 0, 1, 10, 45, 120, 210, 252, 205, 100, 24, 2, 0, 1, 11, 55, 165, 330, 462, 461, 315, 130, 27, 2, 0, 1, 12, 66
Offset: 0
Examples
The square array A(n,k) begins: n\k | 0 1 2 3 4 5 6 7 8 9 10 ====+===================================================== 0 | 1 0 0 0 0 0 0 0 0 0 0 1 | 1 2 2 2 2 2 2 2 2 2 2 2 | 1 3 6 9 12 15 18 21 24 27 30 3 | 1 4 10 20 34 52 74 100 130 164 202 4 | 1 5 15 35 70 125 205 315 460 645 875 5 | 1 6 21 56 126 252 461 786 1266 1946 2877 6 | 1 7 28 84 210 462 924 1715 2996 4977 7924 7 | 1 8 36 120 330 792 1716 3432 6434 11432 19412 8 | 1 9 45 165 495 1287 3003 6435 12870 24309 43749 9 | 1 10 55 220 715 2002 5005 11440 24310 48620 92377 10 | 1 11 66 286 1001 3003 8008 19448 43758 92378 184756 etc. The triangle T(n,k) begins: n\k | 0 1 2 3 4 5 6 7 8 9 10 11 12 ====+============================================== 0 | 1 1 | 1 0 2 | 1 2 0 3 | 1 3 2 0 4 | 1 4 6 2 0 5 | 1 5 10 9 2 0 6 | 1 6 15 20 12 2 0 7 | 1 7 21 35 34 15 2 0 8 | 1 8 28 56 70 52 18 2 0 9 | 1 9 36 84 126 125 74 21 2 0 10 | 1 10 45 120 210 252 205 100 24 2 0 11 | 1 11 55 165 330 462 461 315 130 27 2 0 12 | 1 12 66 220 495 792 924 786 460 164 30 2 0 etc.
Crossrefs
Row sums of the triangle give A099036 for n >= 0.
In the square array; row 0..12 are: A000007, A040000, A008486, A005893, A008487, A008488, A008489, A008490, A008491, A008492, A008493, A008494, A008495.
A173265 is based on the same square array, but is read by descending antidiagonals with special treatment of column 0.
Programs
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GAP
nmax:=15;; A:=List([0..nmax],n->List([0..nmax],k->Binomial(n+k,k)-Binomial(k-1,k-1-n)));; b:=List([2..nmax],n->OrderedPartitions(n,2));; a:=Flat(List([1..Length(b)],i->List([1..Length(b[i])],j->A[b[i][j][2]][b[i][j][1]]))); # Muniru A Asiru, Jul 20 2018
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Mathematica
Table[SeriesCoefficient[(1 - x^(# + 1))/((1 - x)^(# + 1)), {x, 0, k}] &[n - k], {n, 0, 12}, {k, 0, n}] // Flatten (* Michael De Vlieger, Jul 20 2018 *) -
PARI
T(n,k) = binomial(n+k,k) - binomial(k-1,k-1-n); \\ Michel Marcus, Aug 07 2018
Formula
A(n,k) = binomial(n+k,k) - binomial(k-1,k-1-n) for n >= 0 and k >= 0 with binomial(i,j) = 0 if i < j or j < 0.
G.f.: Sum_{k>=0,n>=0} A(n,k)*x^k*y^n = ((1-x)^2)/((1-x-y)*(1-x-x*y)).
Seen as a triangle T(n,k) = A(n-k,k) = binomial(n,k)-binomial(k-1,2*k-1-n) for 0 <= k <= n with binomial(i,j) = 0 if i < j or j < 0.
Mirror image of the triangle equals A173265 except column 0.
Comments