A293304 -id:A293304 - cp's OEIS frontend

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

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

A293461 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^(j(2*i-1))).

Original entry on oeis.org

1, 1, 0, 1, 1, 0, 1, 1, 0, 0, 1, 1, 2, 1, 0, 1, 1, 2, 1, 1, 0, 1, 1, 2, 4, 1, 1, 0, 1, 1, 2, 4, 1, 3, 1, 0, 1, 1, 2, 4, 5, 3, 3, 1, 0, 1, 1, 2, 4, 5, 3, 6, 5, 2, 0, 1, 1, 2, 4, 5, 8, 6, 5, 6, 2, 0, 1, 1, 2, 4, 5, 8, 6, 9, 9, 4, 2, 0, 1, 1, 2, 4, 5, 8, 12, 9, 9, 13

Offset: 0

Seiichi Manyama, Oct 09 2017

			Square array begins:
   1, 1, 1, 1, 1, ...
   0, 1, 1, 1, 1, ...
   0, 0, 2, 2, 2, ...
   0, 1, 1, 4, 4, ...
   0, 1, 1, 1, 5, ...
   0, 1, 3, 3, 3, ...

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

Columns k=0..3 give A000007, A000700, A293304, A293463.
Rows n=0..1 give A000012, A057427.
Main diagonal gives A102186.
Cf. A290216.

max = 12; A[n_, k_] := SeriesCoefficient[Product[(x*(-(k*x^((2*i - 1)*(k + 1) + 1)) - x^((2*i - 1)*(k + 1) + 1) + k*x^((2*i - 1)*(k + 1) + 2*i) + x^(2*i)))/(x^(2*i) - x)^2 + 1, {i, 1, max}], {x, 0, n}]; Flatten[ Table[ A[n - k, k], {n, 0, max}, {k, n, 0, -1}]] (* Jean-François Alcover, Oct 10 2017 *)

A293488 E.g.f.: Product_{m>0} (1 + x^(2m-1) + x^(4m-2)/2!).

Original entry on oeis.org

1, 1, 1, 6, 24, 180, 1080, 10080, 90720, 907200, 10886400, 139708800, 2035756800, 29578348800, 479480601600, 7846046208000, 146459529216000, 2845499424768000, 58421660064768000, 1246862279190528000, 28586598596075520000, 664182248232222720000

Offset: 0

Seiichi Manyama, Oct 10 2017

nonn

Seiichi Manyama, Table of n, a(n) for n = 0..446

Column k=2 of A293486.

Cf. A293138, A293304.

N=66; x='x+O('x^N); Vec(serlaplace(prod(m=1, N, 1+x^(2*m-1)+x^(4*m-2)/2)))

a(n) ~ 2^(-3/4) * c^(1/4) * exp(sqrt(2*c*n) - n) * n^(n-1/4), where c = -polylog(2, -1/2 - I/2) - polylog(2, -1/2 + I/2) = 0.966945612722157030083754546059357521... - Vaclav Kotesovec, Oct 11 2017

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

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A293461 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^(j(2*i-1))).

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A293488 E.g.f.: Product_{m>0} (1 + x^(2m-1) + x^(4m-2)/2!).

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cp's OEIS Frontend

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

Views

Author

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Maple

Mathematica

PARI

A293461 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*(2*i-1))).

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Author

Keywords

Examples

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Crossrefs

Programs

Mathematica

A293488 E.g.f.: Product_{m>0} (1 + x^(2*m-1) + x^(4*m-2)/2!).

Views

Author

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Links

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Programs

PARI

Formula

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

A293461 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^(j(2*i-1))).

A293488 E.g.f.: Product_{m>0} (1 + x^(2m-1) + x^(4m-2)/2!).