A293138 -id:A293138 - cp's OEIS frontend

A293135 Square array A(n,k), n >= 0, k >= 0, read by antidiagonals, where column k is the expansion of e.g.f. Product_{i>0} Sum_{j=0..k} x^(j*i)/j!.

1, 1, 0, 1, 1, 0, 1, 1, 2, 0, 1, 1, 3, 12, 0, 1, 1, 3, 12, 48, 0, 1, 1, 3, 13, 72, 360, 0, 1, 1, 3, 13, 72, 480, 2880, 0, 1, 1, 3, 13, 73, 500, 3780, 25200, 0, 1, 1, 3, 13, 73, 500, 4020, 35280, 241920, 0, 1, 1, 3, 13, 73, 501, 4050, 37380, 372960, 2903040, 0

Offset: 0

Seiichi Manyama, Oct 01 2017

			Square array begins:
   1,   1,   1,   1,   1, ...
   0,   1,   1,   1,   1, ...
   0,   2,   3,   3,   3, ...
   0,  12,  12,  13,  13, ...
   0,  48,  72,  72,  73, ...
   0, 360, 480, 500, 500, ...

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

Columns k=0..5 give A000007, A088311, A293138, A293195, A293196, A293197.
Rows n=0 gives A000012.
Main diagonal gives A000262.
Cf. A293139.

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)-> n!*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_] := n! 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 *)

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

A293141 E.g.f.: Product_{m>0} (1 - x^m + x^(2m)/2).

Original entry on oeis.org

1, -1, -1, 0, 0, 0, 180, 0, 10080, 90720, 0, 0, 179625600, -1556755200, -10897286400, 326918592000, -2615348736000, -88921857024000, 800296713216000, -30411275102208000, 152056375511040000, 0, -351250227430502400000, -3231502092360622080000

Offset: 0

Seiichi Manyama, Oct 01 2017

sign

Seiichi Manyama, Table of n, a(n) for n = 0..449

Column k=2 of A293139.

Cf. A293138.

N:= 30: # to get a(0)..a(N)
P:= mul(1-x^m+x^(2*m)/2, m=1..N):
seq(coeff(P,x,n)*n!, n=0..N); # Robert Israel, Oct 01 2017

A293182 Expansion of Product_{k>=1} (1 + 2x^k - x^(2k)).

Original entry on oeis.org

1, 2, 1, 6, 3, 6, 16, 12, 16, 22, 51, 36, 60, 62, 91, 154, 148, 176, 236, 278, 328, 552, 508, 670, 771, 988, 1068, 1438, 1844, 1998, 2401, 2882, 3300, 4030, 4640, 5406, 7212, 7584, 9072, 10480, 12612, 13964, 17024, 18860, 22545, 27298, 30340, 34372, 41068

Offset: 0

Vaclav Kotesovec, Oct 01 2017

nonn

Robert Israel, Table of n, a(n) for n = 0..10000

Cf. A162891, A263401, A276527, A293138.

N:= 100:
P:= mul(1+2*x^m- x^(2*m), m=1..N):
S:= series(P,x,N+1):
seq(coeff(S,x,n), n=0..N); # Robert Israel, Oct 01 2017

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

a(n) ~ c^(1/4) * exp(2*sqrt(c*n)) / (2^(3/2) * sqrt(Pi) * n^(3/4)), where c = Pi^2/6 + log(1+sqrt(2))^2/2 + polylog(2, 3-2*sqrt(2))/2 - 2*polylog(2, sqrt(2)-1) = 1.18805291660775259061867850175092520191179528961165451864292...

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

cp's OEIS Frontend

A293135 Square array A(n,k), n >= 0, k >= 0, read by antidiagonals, where column k is the expansion of e.g.f. Product_{i>0} Sum_{j=0..k} x^(j*i)/j!.

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

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A293141 E.g.f.: Product_{m>0} (1 - x^m + x^(2m)/2).

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A293182 Expansion of Product_{k>=1} (1 + 2*x^k - x^(2*k)).

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Maple

Mathematica

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

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

A293182 Expansion of Product_{k>=1} (1 + 2x^k - x^(2k)).

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