A287696 -id:A287696 - cp's OEIS frontend

A287698 Square array A(n,k) = (n!)^3 [x^n] hypergeom([], [1, 1], z)^k read by antidiagonals.

1, 1, 0, 1, 1, 0, 1, 2, 1, 0, 1, 3, 10, 1, 0, 1, 4, 27, 56, 1, 0, 1, 5, 52, 381, 346, 1, 0, 1, 6, 85, 1192, 6219, 2252, 1, 0, 1, 7, 126, 2705, 36628, 111753, 15184, 1, 0, 1, 8, 175, 5136, 124405, 1297504, 2151549, 104960, 1, 0

Offset: 0

Peter Luschny, May 30 2017

Let A_m(n,k) = (n!)^m [x^n] hypergeom([], [1,…,1], z)^k where [1,…,1] lists (m-1) times 1. These arrays can be seen as generalizations of the power functions n^k. For m = 1 -> A003992, m = 2 -> A287316, m = 3 -> A287698.
A_m(n,n) is the sum of m-th powers of coefficients in the full expansion of (z_1+z_2+...+z_n)^n (compare A245397).
A287696 provide polynomials and A287697 rational functions generating the columns of the array.

			Array starts:
k\n| 0  1    2       3       4         5           6             7
---|-------------------------------------------------------------------
k=0| 1, 0,   0,      0,      0,        0,          0,            0, ... A000007
k=1| 1, 1,   1,      1,      1,        1,          1,            1, ... A000012
k=2| 1, 2,  10,     56,    346,     2252,      15184,       104960, ... A000172
k=3| 1, 3,  27,    381,   6219,   111753,    2151549,     43497891, ... A141057
k=4| 1, 4,  52,   1192,  36628,  1297504,   50419096,   2099649808, ... A287699
k=5| 1, 5,  85,   2705, 124405,  7120505,  464011825,  33031599725, ...
k=6| 1, 6, 126,   5136, 316206, 25461756, 2443835736, 263581282656, ...
       A001107,A287702,A287700,  A287701,                               A055733

Rows: A000007 (k=0), A000012 (k=1), A000172 (k=2), A141057 (k=3), A287699 (k=4).
Columns: A000172 (n=1), A001477(n=1), A001107 (n=2), A287702 (n=3), A287700 (n=4), A287701 (n=5).
Cf. A055733 (diagonal), A287696, A287697, A003992, A287316, A245397.

A287698_row := (k, len) -> seq(A287696_poly(j)(k), j=0..len):
A287698_row := proc(k, len) hypergeom([], [1, 1], x):
series(%^k, x, len); seq((i!)^3*coeff(%, x, i), i=0..len-1) end:
for k from 0 to 6 do A287698_row(k, 9) od;
A287698_col := proc(n, len) local k, x; hypergeom([], [1, 1], z);
series(%^x, z=0, n+1): unapply(n!^3*coeff(%, z, n), x); seq(%(j), j=0..len) end:
for n from 0 to 7 do A287698_col(n, 9) od;

Table[Table[SeriesCoefficient[HypergeometricPFQ[{},{1,1},x]^k, {x, 0, n}] (n!)^3, {n, 0, 6}], {k, 0, 9}] (* as a table of rows *)

A287697 Triangle read by rows, (Sum_{k=0..n} T[n,k]*x^k) / (1-x)^(n+1) are generating functions of the columns of A287698.

Original entry on oeis.org

1, 0, 1, 0, 1, 7, 0, 1, 52, 163, 0, 1, 341, 4499, 8983, 0, 1, 2246, 98256, 660746, 966751, 0, 1, 15177, 2045282, 35677082, 155729277, 179781181, 0, 1, 104952, 42658239, 1754605504, 17446464519, 55690144728, 53090086057

Offset: 0

Peter Luschny, May 30 2017

			Triangle starts:
0: [1]
1: [0, 1]
2: [0, 1,      7]
3: [0, 1,     52,      163]
4: [0, 1,    341,     4499,       8983]
5: [0, 1,   2246,    98256,     660746,      966751]
6: [0, 1,  15177,  2045282,   35677082,   155729277,   179781181]
7: [0, 1, 104952, 42658239, 1754605504, 17446464519, 55690144728, 53090086057]
...
Let q4(x) = (x + 341*x^2 + 4499*x^3 + 8983*x^4) / (1-x)^5 then the coefficients of the series expansion of q4 are column 4 of A287698.

Cf. A000442, A212856, A287696, A287698.

A287697_row := n -> Delta(A287696_poly(n), n): # Delta defined in A287315.
for n from 0 to 9 do A287697_row(n) od;
A287697_eulerian := (n,x) -> add(A287697_row(n)[k+1]*x^k,k=0..n)/(1-x)^(n+1):
for n from 0 to 4 do A287697_eulerian(n,x) od;

T(n,n) = A212856(n).

Sum_{k=0..n} T(n,k) = A000442(n).

cp's OEIS Frontend

A287698 Square array A(n,k) = (n!)^3 [x^n] hypergeom([], [1, 1], z)^k read by antidiagonals.

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