A293204 -id:A293204 - cp's OEIS frontend

A293202 Square array A(n,k), n >= 0, k >= 0, read by antidiagonals, where column k is the expansion of g.f. Product_{i>0} Sum_{j=0..k} j!x^(ji).

1, 1, 0, 1, 1, 0, 1, 1, 1, 0, 1, 1, 3, 2, 0, 1, 1, 3, 2, 2, 0, 1, 1, 3, 8, 6, 3, 0, 1, 1, 3, 8, 6, 7, 4, 0, 1, 1, 3, 8, 30, 13, 12, 5, 0, 1, 1, 3, 8, 30, 13, 24, 13, 6, 0, 1, 1, 3, 8, 30, 133, 48, 37, 22, 8, 0, 1, 1, 3, 8, 30, 133, 48, 61, 46, 26, 10, 0, 1, 1, 3, 8, 30, 133, 768, 181, 142, 98, 42, 12, 0

Offset: 0

Seiichi Manyama, Oct 02 2017

			Square array begins:
   1, 1, 1,  1,  1, ...
   0, 1, 1,  1,  1, ...
   0, 1, 3,  3,  3, ...
   0, 2, 2,  8,  8, ...
   0, 2, 6,  6, 30, ...
   0, 3, 7, 13, 13, ...

Seiichi Manyama, Antidiagonals n = 0..139, flattened

Columns k=0..5 give A000007, A000009, A293204, A289485, A289486, A293250.
Rows n=0..1 give A000012, A057427.
Main diagonal gives A161779.

b:= proc(n, i, k) option remember; `if`(n=0, 1, `if`(i<1, 0,
      add(b(n-i*j, i-1, k)*j!, j=0..min(k, n/i))))
    end:
A:= (n, k)-> b(n$2, k):
seq(seq(A(n, d-n), n=0..d), d=0..12);  # Alois P. Heinz, Oct 02 2017

b[n_, i_, k_] := b[n, i, k] = If[n == 0, 1, If[i < 1, 0, Sum[b[n - i j, i - 1, k] j!, {j, 0, Min[k, n/i]}]]];
A [n_, k_] := b[n, n, k];
Table[Table[A[n, d-n], {n, 0, d}], {d, 0, 12}] // Flatten (* Jean-François Alcover, Dec 06 2019, after Alois P. Heinz *)

A290216 Square array A(n,k), n >= 0, k >= 0, read by antidiagonals, where column k is the expansion of g.f. Product_{i>0} (1 + Sum_{j=1..k} jx^(ji)).

Original entry on oeis.org

1, 1, 0, 1, 1, 0, 1, 1, 1, 0, 1, 1, 3, 2, 0, 1, 1, 3, 2, 2, 0, 1, 1, 3, 5, 6, 3, 0, 1, 1, 3, 5, 6, 7, 4, 0, 1, 1, 3, 5, 10, 10, 12, 5, 0, 1, 1, 3, 5, 10, 10, 18, 13, 6, 0, 1, 1, 3, 5, 10, 15, 22, 25, 22, 8, 0, 1, 1, 3, 5, 10, 15, 22, 29, 34, 26, 10, 0, 1, 1, 3, 5

Offset: 0

Seiichi Manyama, Oct 06 2017

			Square array begins:
   1, 1, 1,  1,  1, ...
   0, 1, 1,  1,  1, ...
   0, 1, 3,  3,  3, ...
   0, 2, 2,  5,  5, ...
   0, 2, 6,  6, 10, ...
   0, 3, 7, 10, 10, ...

Seiichi Manyama, Antidiagonals n = 0..139, flattened

Columns k=0..3 give A000007, A000009, A293204, A290269.
Rows n=0 gives A000012.
Main diagonal gives A077285.
Product_{i>0} (1 + Sum_{j=1..k} j*x^(j*i))^m: A290217 (m=-1), this sequence (m=1), A293377 (m=2).
Cf. A293305.

A293072 G.f.: Product_{m>0} (1-x^m+2!x^(2m)).

Original entry on oeis.org

1, -1, 1, 0, 0, -3, 4, -3, 4, -2, 0, -6, 9, -10, 10, -5, 10, -16, 20, -28, 22, -18, 19, -32, 54, -50, 45, -38, 44, -78, 106, -118, 96, -98, 110, -129, 192, -216, 204, -182, 213, -286, 368, -412, 366, -362, 394, -524, 676, -714, 680, -641, 742, -936, 1170

Offset: 0

Seiichi Manyama, Oct 03 2017

sign

Alois P. Heinz, Table of n, a(n) for n = 0..10000 (first 501 terms from Seiichi Manyama)

Column k=2 of A293071.

Cf. A293204, A293304.

b:= proc(n, i) option remember; `if`(n=0, 1, `if`(i<1, 0,
      add(b(n-i*j, i-1)*j!*(-1)^j, j=0..min(2, n/i))))
    end:
a:= n-> b(n$2):
seq(a(n), n=0..60);  # Alois P. Heinz, Oct 04 2017

nmax = 100; CoefficientList[Series[Product[1 - x^k + 2*x^(2*k), {k, 1, nmax}], {x, 0, nmax}], x] (* Vaclav Kotesovec, Oct 05 2017 *)

Vec(prod(m=1, 80, 1-x^m+2*x^(2*m)) + O(x^80)) \\ Michel Marcus, Oct 04 2017

A293287 G.f.: Product_{m>0} 1/(1 + x^m + 2!x^(2m)).

Original entry on oeis.org

1, -1, -2, 3, -1, -5, 9, 3, -20, 16, 20, -54, 19, 89, -123, -56, 287, -181, -383, 748, 37, -1520, 1406, 1619, -4439, 1190, 7768, -10102, -5494, 25673, -14754, -36660, 66367, 7047, -139852, 125785, 153573, -405349, 98671, 712275, -909382, -514965, 2332540, -1303341

Offset: 0

Seiichi Manyama, Oct 04 2017

sign

Seiichi Manyama, Table of n, a(n) for n = 0..500

Column k=2 of A293285.

Cf. A293204.

Vec(prod(m=1, 50, 1/(1 + x^m + 2!*x^(2*m))) + O(x^50)) \\ Michel Marcus, Oct 05 2017

Convolution inverse of A293204.

A293253 Expansion of Product_{k>=1} (1 + x^k + x^(k^2)).

Original entry on oeis.org

1, 2, 1, 3, 4, 6, 6, 8, 12, 15, 20, 22, 30, 35, 46, 53, 67, 80, 97, 117, 138, 165, 195, 231, 272, 323, 378, 442, 514, 600, 696, 806, 931, 1078, 1240, 1431, 1638, 1881, 2147, 2461, 2802, 3197, 3632, 4131, 4685, 5310, 6009, 6790, 7670, 8652, 9749, 10968, 12336

Offset: 0

Vaclav Kotesovec, Oct 04 2017

nonn

Vaclav Kotesovec, Table of n, a(n) for n = 0..10000

Cf. A033461, A263401, A293204, A293254.

nmax = 100; CoefficientList[Series[Product[(1+x^k+x^(k^2)), {k, 1, nmax}], {x, 0, nmax}], x]

cp's OEIS Frontend

A293202 Square array A(n,k), n >= 0, k >= 0, read by antidiagonals, where column k is the expansion of g.f. Product_{i>0} Sum_{j=0..k} j!*x^(j*i).

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Maple

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A290216 Square array A(n,k), n >= 0, k >= 0, read by antidiagonals, where column k is the expansion of g.f. Product_{i>0} (1 + Sum_{j=1..k} j*x^(j*i)).

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A293072 G.f.: Product_{m>0} (1-x^m+2!*x^(2*m)).

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Maple

Mathematica

PARI

A293287 G.f.: Product_{m>0} 1/(1 + x^m + 2!*x^(2*m)).

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PARI

Formula

A293253 Expansion of Product_{k>=1} (1 + x^k + x^(k^2)).

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Author

Keywords

Links

Crossrefs

Programs

Mathematica

A293202 Square array A(n,k), n >= 0, k >= 0, read by antidiagonals, where column k is the expansion of g.f. Product_{i>0} Sum_{j=0..k} j!x^(ji).

A290216 Square array A(n,k), n >= 0, k >= 0, read by antidiagonals, where column k is the expansion of g.f. Product_{i>0} (1 + Sum_{j=1..k} jx^(ji)).

A293072 G.f.: Product_{m>0} (1-x^m+2!x^(2m)).

A293287 G.f.: Product_{m>0} 1/(1 + x^m + 2!x^(2m)).