A293202 -id:A293202 - 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

A289485 G.f.: Product_{m>0} (1 + x^m + 2!x^(2m) + 3!x^(3m)).

Original entry on oeis.org

1, 1, 3, 8, 6, 13, 24, 37, 46, 98, 90, 142, 235, 296, 392, 601, 746, 1018, 1428, 1702, 2258, 3132, 3891, 5044, 6766, 8134, 10681, 13812, 16952, 20996, 27986, 33144, 41376, 52500, 64066, 79401, 99718, 119590, 147162, 181880, 220339, 266662, 329342, 391736

Offset: 0

Seiichi Manyama, Oct 03 2017

nonn

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

Column k=3 of A293202.

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(3, n/i))))
    end:
a:= n-> b(n$2):
seq(a(n), n=0..50);  # Alois P. Heinz, Oct 04 2017

Vec(prod(m=1, 50, 1+x^m+2*x^(2*m)+6*x^(3*m)) + O(x^50)) \\ Michel Marcus, Oct 04 2017

A289486 G.f.: Product_{m>0} (1 + x^m + 2!x^(2m) + 3!x^(3m) + 4!x^(4m)).

Original entry on oeis.org

1, 1, 3, 8, 30, 13, 48, 61, 142, 170, 378, 430, 1099, 872, 1304, 2305, 3626, 4042, 6900, 8086, 13634, 17628, 24915, 29524, 53998, 51718, 74953, 96996, 150104, 172292, 234938, 296472, 416064, 504132, 690826, 812625, 1184134, 1378774, 1823778, 2264576, 3115051

Offset: 0

Seiichi Manyama, Oct 03 2017

nonn

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

Column k=4 of A293202.

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(4, n/i))))
    end:
a:= n-> b(n$2):
seq(a(n), n=0..50);  # Alois P. Heinz, Oct 04 2017

A293250 G.f.: Product_{m>0} (1 + x^m + 2!x^(2m) + 3!x^(3m) + 4!x^(4m) + 5!x^(5m)).

Original entry on oeis.org

1, 1, 3, 8, 30, 133, 48, 181, 262, 530, 738, 1750, 1939, 5672, 6704, 20065, 11306, 20362, 36780, 53926, 77234, 116988, 146955, 219604, 353518, 479638, 643753, 1156116, 1393784, 1778612, 3767018, 3224712, 4227504, 6058692, 9406426, 13136745, 15425494, 19723654

Offset: 0

Seiichi Manyama, Oct 03 2017

nonn

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

Column k=5 of A293202.

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(5, n/i))))
    end:
a:= n-> b(n$2):
seq(a(n), n=0..50);  # Alois P. Heinz, Oct 04 2017

A293285 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=0..k} j!x^(ji)).

Original entry on oeis.org

1, 1, 0, 1, -1, 0, 1, -1, 0, 0, 1, -1, -2, -1, 0, 1, -1, -2, 3, 1, 0, 1, -1, -2, -3, -1, -1, 0, 1, -1, -2, -3, 11, -5, 1, 0, 1, -1, -2, -3, -13, 7, 9, -1, 0, 1, -1, -2, -3, -13, 55, -15, 3, 2, 0, 1, -1, -2, -3, -13, -65, 33, -63, -20, -2, 0, 1, -1, -2, -3, -13, -65

Offset: 0

Seiichi Manyama, Oct 04 2017

			Square array begins:
   1,  1,  1,  1,   1, ...
   0, -1, -1, -1,  -1, ...
   0,  0, -2, -2,  -2, ...
   0, -1,  3, -3,  -3, ...
   0,  1, -1, 11, -13, ...
   0, -1, -5,  7,  55, ...

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

Columns k=0..2 give A000007, A081362, A293287.
Rows n=0..1 give A000012, (-1)*A057427.
Main diagonal gives A293251.
Cf. A293202, A293293.

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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Mathematica

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

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A289485 G.f.: Product_{m>0} (1 + x^m + 2!*x^(2*m) + 3!*x^(3*m)).

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PARI

A289486 G.f.: Product_{m>0} (1 + x^m + 2!*x^(2*m) + 3!*x^(3*m) + 4!*x^(4*m)).

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A293250 G.f.: Product_{m>0} (1 + x^m + 2!*x^(2*m) + 3!*x^(3*m) + 4!*x^(4*m) + 5!*x^(5*m)).

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Maple

A293285 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=0..k} j!*x^(j*i)).

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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)).

A289485 G.f.: Product_{m>0} (1 + x^m + 2!x^(2m) + 3!x^(3m)).

A289486 G.f.: Product_{m>0} (1 + x^m + 2!x^(2m) + 3!x^(3m) + 4!x^(4m)).

A293250 G.f.: Product_{m>0} (1 + x^m + 2!x^(2m) + 3!x^(3m) + 4!x^(4m) + 5!x^(5m)).

A293285 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=0..k} j!x^(ji)).