A293072 -id:A293072 - cp's OEIS frontend

A293071 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} (-1)^jj!x^(j*i).

1, 1, 0, 1, -1, 0, 1, -1, -1, 0, 1, -1, 1, 0, 0, 1, -1, 1, 0, 0, 0, 1, -1, 1, -6, 0, 1, 0, 1, -1, 1, -6, 0, -3, 0, 0, 1, -1, 1, -6, 24, 3, 4, 1, 0, 1, -1, 1, -6, 24, 3, 4, -3, 0, 0, 1, -1, 1, -6, 24, -117, -20, -3, 4, 0, 0, 1, -1, 1, -6, 24, -117, -20, -27, -8, -2

Offset: 0

Seiichi Manyama, Oct 03 2017

			Square array begins:
   1,  1,  1,  1,  1, ...
   0, -1, -1, -1, -1, ...
   0, -1,  1,  1,  1, ...
   0,  0,  0, -6, -6, ...
   0,  0,  0,  0, 24, ...
   0,  1, -3,  3,  3, ...

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

Columns k=0..5 give A000007, A010815, A293072, A293255, A293256, A293257.
Rows n=0 gives A000012.
Main diagonal gives A293236.
Cf. A293202.

nmax = 12;
col[k_] := CoefficientList[Product[Sum[(-1)^j j! x^(i j), {j, 0, k}], {i, 1, nmax+1}] + O[x]^(nmax+1), x]; M = PadRight[col[#], nmax+1]& /@ Range[0, nmax] // Transpose;
A[n_, k_] := M[[n+1, k+1]];
Table[A[n-k, k], {n, 0, nmax}, {k, n, 0, -1}] // Flatten (* Jean-François Alcover, Nov 15 2020 *)

A293204 G.f.: Product_{m>0} (1+x^m+2!x^(2m)).

Original entry on oeis.org

1, 1, 3, 2, 6, 7, 12, 13, 22, 26, 42, 46, 73, 80, 116, 139, 194, 226, 306, 358, 482, 558, 735, 856, 1108, 1300, 1657, 1926, 2426, 2834, 3530, 4110, 5082, 5898, 7234, 8409, 10216, 11860, 14304, 16568, 19891, 22990, 27470, 31670, 37630, 43382, 51274, 58982, 69450

Offset: 0

Seiichi Manyama, Oct 02 2017

nonn

			Let's consider the partitions of n where no positive integer appears more than twice. (See A000726)
For n = 5,
    partition      |                         |
--------------------------------------------------------------
     5             -> one 5                  -> 1!       (= 1)
   = 4 + 1         -> one 4 and one 1        -> 1!*1!    (= 1)
   = 3 + 2         -> one 3 and one 2        -> 1!*1!    (= 1)
   = 3 + 1 + 1     -> one 3 and two 1        -> 1!*2!    (= 2)
   = 2 + 2 + 1     -> two 2 and one 1        -> 2!*1!    (= 2)
--------------------------------------------------------------
                                                a(5)      = 7.
For n = 6,
    partition      |                         |
--------------------------------------------------------------
     6             -> one 6                  -> 1!       (= 1)
   = 5 + 1         -> one 5 and one 1        -> 1!*1!    (= 1)
   = 4 + 2         -> one 4 and one 2        -> 1!*1!    (= 1)
   = 4 + 1 + 1     -> one 4 and two 1        -> 1!*2!    (= 2)
   = 3 + 3         -> two 3                  -> 2!       (= 2)
   = 3 + 2 + 1     -> one 3, one 2 and one 1 -> 1!*1!*1! (= 1)
   = 2 + 2 + 1 + 1 -> two 2 and two 1        -> 2!*2!    (= 4)
--------------------------------------------------------------
                                                a(6)      = 12.

Alois P. Heinz, Table of n, a(n) for n = 0..5000 (first 501 terms from Seiichi Manyama)

Column k=2 of A293202.
Cf. A000726, A263401, A293138, A293182.
Cf. A293072.

b:= proc(n, i) option remember; `if`(n=0, 1, `if`(i<1, 0,
      add(b(n-i*j, i-1)*j!, j=0..min(2, n/i))))
    end:
a:= n-> b(n$2):
seq(a(n), n=0..50);  # Alois P. Heinz, Oct 02 2017

nmax = 100; CoefficientList[Series[Product[1 + x^k + 2*x^(2*k), {k, 1, nmax}], {x, 0, nmax}], x] (* Vaclav Kotesovec, Oct 02 2017 *)

a(n) ~ c^(1/4) * exp(2*sqrt(c*n)) / (4 * sqrt(Pi) * n^(3/4)), where c = Pi^2/3 - arctan(sqrt(7))^2 + log(2)^2/4 + polylog(2, -1/4 - I*sqrt(7)/4) + polylog(2, -1/4 + I*sqrt(7)/4) = 1.323865936864425754643630663383779192757247984691212163137... - Vaclav Kotesovec, Oct 02 2017

Equivalently, c = -polylog(2, -1/2 + I*sqrt(7)/2) - polylog(2, -1/2 - I*sqrt(7)/2). - Vaclav Kotesovec, Oct 05 2017

A293305 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} (-1)^jjx^(j*i)).

Original entry on oeis.org

1, 1, 0, 1, -1, 0, 1, -1, -1, 0, 1, -1, 1, 0, 0, 1, -1, 1, 0, 0, 0, 1, -1, 1, -3, 0, 1, 0, 1, -1, 1, -3, 0, -3, 0, 0, 1, -1, 1, -3, 4, 0, 4, 1, 0, 1, -1, 1, -3, 4, 0, 4, -3, 0, 0, 1, -1, 1, -3, 4, -5, 0, -3, 4, 0, 0, 1, -1, 1, -3, 4, -5, 0, -7, -2, -2, 0, 0, 1, -1, 1

Offset: 0

Seiichi Manyama, Oct 05 2017

			Square array begins:
   1,  1,  1,  1,  1, ...
   0, -1, -1, -1, -1, ...
   0, -1,  1,  1,  1, ...
   0,  0,  0, -3, -3, ...
   0,  0,  0,  0,  4, ...
   0,  1, -3,  0,  0, ...

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

Columns k=0..2 give A000007, A010815, A293072.
Rows n=0 gives A000012.
Main diagonal gives A293306.
Cf. A293071, A293307.

nmax = 12;
col[k_] := col[k] = Product[1+Sum[(-1)^j*j*x^(i*j), {j, 1, k}], {i, 1, 2 nmax}] + O[x]^(2 nmax) // CoefficientList[#, x]&;
A[n_, k_] := If[n == 0, 1, If[k == 0, 0, col[k][[n+1]]]];
Table[A[n-k, k], {n, 0, nmax}, {k, n, 0, -1}] // Flatten (* Jean-François Alcover, Feb 21 2021 *)

A293304 Expansion of Product_{k>=1} (1 + x^(2k-1) + 2x^(4*k-2)).

Original entry on oeis.org

1, 1, 2, 1, 1, 3, 3, 5, 6, 4, 6, 8, 11, 13, 13, 18, 19, 23, 29, 32, 35, 40, 48, 51, 65, 78, 86, 96, 102, 121, 142, 162, 179, 199, 220, 251, 289, 323, 359, 395, 450, 499, 562, 631, 695, 762, 840, 952, 1055, 1167, 1292, 1413, 1557, 1733, 1903, 2112, 2323, 2534

Offset: 0

Vaclav Kotesovec, Oct 05 2017

nonn

Vaclav Kotesovec, Table of n, a(n) for n = 0..2000

Cf. A293072.

nmax = 100; CoefficientList[Series[Product[(1 + x^(2*k-1) + 2*x^(4*k-2)), {k, 1, nmax}], {x, 0, nmax}], x]

a(n) ~ c^(1/4) * exp(sqrt(2*c*n)) / (2^(5/4) * sqrt(Pi) * n^(3/4)), where c = -polylog(2, -1/2 + I*sqrt(7)/2) - polylog(2, -1/2 - I*sqrt(7)/2) = 1.323865936864425754643630663383779192757247984691212163137...

A293294 G.f.: Product_{m>0} 1/(1 - x^m + 2!x^(2m)).

Original entry on oeis.org

1, 1, 0, -1, -1, 3, 3, -1, -8, -8, 12, 24, 5, -43, -55, 40, 137, 65, -215, -356, 97, 780, 624, -941, -2199, -260, 4052, 4638, -3536, -12861, -5676, 19858, 31449, -8337, -71220, -54243, 87733, 196679, 20733, -372807, -413794, 330731, 1159718, 497517, -1821469

Offset: 0

Seiichi Manyama, Oct 05 2017

sign

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

Column k=2 of A293293.

Cf. A293072.

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

Convolution inverse of A293072.

cp's OEIS Frontend

A293071 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} (-1)^j*j!*x^(j*i).

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A293204 G.f.: Product_{m>0} (1+x^m+2!*x^(2*m)).

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A293305 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} (-1)^j*j*x^(j*i)).

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Mathematica

A293304 Expansion of Product_{k>=1} (1 + x^(2*k-1) + 2*x^(4*k-2)).

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Mathematica

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A293294 G.f.: Product_{m>0} 1/(1 - x^m + 2!*x^(2*m)).

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A293071 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} (-1)^jj!x^(j*i).

A293204 G.f.: Product_{m>0} (1+x^m+2!x^(2m)).

A293305 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} (-1)^jjx^(j*i)).

A293304 Expansion of Product_{k>=1} (1 + x^(2k-1) + 2x^(4*k-2)).

A293294 G.f.: Product_{m>0} 1/(1 - x^m + 2!x^(2m)).