A293135 -id:A293135 - cp's OEIS frontend

A293138 E.g.f.: Product_{m>0} (1+x^m+x^(2*m)/2!).

1, 1, 3, 12, 72, 480, 3780, 35280, 372960, 4263840, 54432000, 758419200, 11436163200, 185253868800, 3214699488000, 59172265152000, 1163830187520000, 24097823253504000, 525794940582912000, 12073276215576576000, 290883846352619520000, 7318777466097377280000

Offset: 0

Seiichi Manyama, Oct 01 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!)       (= 1  )
   = 4 + 1         -> one 4 and one 1        -> 1/(1!*1!)    (= 1  )
   = 3 + 2         -> one 3 and one 2        -> 1/(1!*1!)    (= 1  )
   = 3 + 1 + 1     -> one 3 and two 1        -> 1/(1!*2!)    (= 1/2)
   = 2 + 2 + 1     -> two 2 and one 1        -> 1/(2!*1!)    (= 1/2)
--------------------------------------------------------------------
                                                sum             4
So a(5) = 5! * 4 = 480.
For n = 6,
    partition      |                         |
--------------------------------------------------------------------
     6             -> one 6                  -> 1/(1!)       (= 1  )
   = 5 + 1         -> one 5 and one 1        -> 1/(1!*1!)    (= 1  )
   = 4 + 2         -> one 4 and one 2        -> 1/(1!*1!)    (= 1  )
   = 4 + 1 + 1     -> one 4 and two 1        -> 1/(1!*2!)    (= 1/2)
   = 3 + 3         -> two 3                  -> 1/(2!)       (= 1/2)
   = 3 + 2 + 1     -> one 3, one 2 and one 1 -> 1/(1!*1!*1!) (= 1  )
   = 2 + 2 + 1 + 1 -> two 2 and two 1        -> 1/(2!*2!)    (= 1/4)
--------------------------------------------------------------------
                                                sum            21/4
So a(6) = 6! * 21/4 = 3780.

Seiichi Manyama, Table of n, a(n) for n = 0..444

Column k=2 of A293135.

Cf. A000726, A162891, A263401, A293141.

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-> n!*b(n$2):
seq(a(n), n=0..23);  # Alois P. Heinz, Oct 02 2017

b[n_, i_] := b[n, i] = If[n == 0, 1, If[i < 1, 0, Sum[b[n - i j, i - 1]/j!, {j, 0, Min[2, n/i]}]]];
a[n_] := n! b[n, n];
a /@ Range[0, 23] (* Jean-François Alcover, Nov 01 2020, after Alois P. Heinz *)

a(n) ~ c^(1/4) * exp(2*sqrt(c*n) - n) * n^(n+1/2) / (sqrt(5) * n^(3/4)), where c = -polylog(2, -1/2 - I/2) - polylog(2, -1/2 + I/2) = 0.9669456127221570300837545... Equivalently, c = -Sum_{k>=1} (-1)^k * cos(Pi*k/4) / (k^2 * 2^(k/2-1)). - Vaclav Kotesovec, Oct 01 2017

A293139 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} (-1)^jx^(ji)/j!.

Original entry on oeis.org

1, 1, 0, 1, -1, 0, 1, -1, -2, 0, 1, -1, -1, 0, 0, 1, -1, -1, 0, 0, 0, 1, -1, -1, -1, 0, 120, 0, 1, -1, -1, -1, 0, 0, 0, 0, 1, -1, -1, -1, 1, 20, 180, 5040, 0, 1, -1, -1, -1, 1, 20, 180, 0, 0, 0, 1, -1, -1, -1, 1, 19, 150, 1260, 10080, 0, 0, 1, -1, -1, -1, 1, 19

Offset: 0

Seiichi Manyama, Oct 01 2017

			Square array begins:
   1,   1,  1,  1,  1, ...
   0,  -1, -1, -1, -1, ...
   0,  -2, -1, -1, -1, ...
   0,   0,  0, -1, -1, ...
   0,   0,  0,  0,  1, ...
   0, 120,  0, 20, 20, ...

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

Columns k=0..2 give A000007, A293140, A293141.
Rows n=0 gives A000012.
Main diagonal gives A293116.
Cf. A293135.

A293486 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(2i-1))/j!.

Original entry on oeis.org

1, 1, 0, 1, 1, 0, 1, 1, 0, 0, 1, 1, 1, 6, 0, 1, 1, 1, 6, 24, 0, 1, 1, 1, 7, 24, 120, 0, 1, 1, 1, 7, 24, 180, 720, 0, 1, 1, 1, 7, 25, 180, 1080, 5040, 0, 1, 1, 1, 7, 25, 180, 1200, 10080, 80640, 0, 1, 1, 1, 7, 25, 181, 1200, 10080, 90720, 725760, 0, 1, 1, 1, 7, 25

Offset: 0

Seiichi Manyama, Oct 10 2017

			Square array begins:
   1,   1,   1,   1,   1, ...
   0,   1,   1,   1,   1, ...
   0,   0,   1,   1,   1, ...
   0,   6,   6,   7,   7, ...
   0,  24,  24,  24,  25, ...
   0, 120, 180, 180, 180, ...

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

Columns k=0..3 give A000007, A293487, A293488, A293489.
Rows n=0 gives A000012.
Main diagonal gives A088009.
Cf. A293135.

A293299 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} 1/(Sum_{j=0..k} x^(j*i)/j!).

Original entry on oeis.org

1, 1, 0, 1, -1, 0, 1, -1, 0, 0, 1, -1, -1, -6, 0, 1, -1, -1, 0, 24, 0, 1, -1, -1, -1, -6, -120, 0, 1, -1, -1, -1, 2, 30, 720, 0, 1, -1, -1, -1, 1, 10, 270, -5040, 0, 1, -1, -1, -1, 1, 20, 170, 0, 80640, 0, 1, -1, -1, -1, 1, 19, 140, 1050, 2520, -725760, 0, 1, -1

Offset: 0

Seiichi Manyama, Oct 05 2017

			Square array begins:
   1,    1,  1,  1,  1, ...
   0,   -1, -1, -1, -1, ...
   0,    0, -1, -1, -1, ...
   0,   -6,  0, -1, -1, ...
   0,   24, -6,  2,  1, ...
   0, -120, 30, 10, 20, ...

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

Columns k=0..1 give A000007, A293300.
Rows n=0 gives A000012.
Main diagonal gives A293116.
Cf. A293135, A293301.

A293301 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} 1/(Sum_{j=0..k} (-1)^jx^(ji)/j!).

Original entry on oeis.org

1, 1, 0, 1, 1, 0, 1, 1, 4, 0, 1, 1, 3, 18, 0, 1, 1, 3, 12, 120, 0, 1, 1, 3, 13, 66, 840, 0, 1, 1, 3, 13, 74, 450, 7920, 0, 1, 1, 3, 13, 73, 510, 3510, 75600, 0, 1, 1, 3, 13, 73, 500, 4130, 32760, 887040, 0, 1, 1, 3, 13, 73, 501, 4040, 38430, 335160, 10886400, 0, 1, 1, 3

Offset: 0

Seiichi Manyama, Oct 05 2017

			Square array begins:
   1,   1,   1,   1,   1, ...
   0,   1,   1,   1,   1, ...
   0,   4,   3,   3,   3, ...
   0,  18,  12,  13,  13, ...
   0, 120,  66,  74,  73, ...
   0, 840, 450, 510, 500, ...

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

Columns k=0..2 give A000007, A053529, A293302.
Rows n=0 gives A000012.
Main diagonal gives A000262.
Cf. A293135, A293139, A293299.

A293195 E.g.f.: Product_{m>0} (1+x^m+x^(2m)/2!+x^(3m)/3!).

Original entry on oeis.org

1, 1, 3, 13, 72, 500, 4020, 37380, 389760, 4546080, 58363200, 814968000, 12301027200, 200216016000, 3484710028800, 64639070496000, 1270187702784000, 26414731639296000, 578733086131200000, 13328586071184384000, 321801976039864320000, 8127599117746268160000

Offset: 0

Seiichi Manyama, Oct 02 2017

nonn

Seiichi Manyama, Table of n, a(n) for n = 0..444

Column k=3 of A293135.

Cf. A001935.

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-> n!*b(n$2):
seq(a(n), n=0..23);  # Alois P. Heinz, Oct 02 2017

A293196 E.g.f.: Product_{m>0} (1+x^m+x^(2m)/2!+x^(3m)/3!+x^(4*m)/4!).

Original entry on oeis.org

1, 1, 3, 13, 73, 500, 4050, 37590, 393960, 4591440, 58842000, 822729600, 12451546800, 202637635200, 3528601876800, 65440475100000, 1287768658176000, 26788326941376000, 587341833414336000, 13533158213422848000, 326898680478289920000, 8260278864850172160000

Offset: 0

Seiichi Manyama, Oct 02 2017

nonn

Seiichi Manyama, Table of n, a(n) for n = 0..444

Column k=4 of A293135.

A293197 E.g.f.: Product_{m>0} (1+x^m+x^(2m)/2!+x^(3m)/3!+x^(4m)/4!+x^(5m)/5!).

Original entry on oeis.org

1, 1, 3, 13, 73, 501, 4050, 37632, 394296, 4595976, 58932720, 823949280, 12467513520, 202933851120, 3534322952160, 65559527953920, 1290129736896000, 26840074522060800, 588476031700677120, 13560642403296468480, 327603687047488051200, 8278733693718654566400

Offset: 0

Seiichi Manyama, Oct 02 2017

nonn

Seiichi Manyama, Table of n, a(n) for n = 0..444

Column k=5 of A293135.

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-> n!*b(n$2):
seq(a(n), n=0..23);  # Alois P. Heinz, Oct 02 2017

cp's OEIS Frontend

A293138 E.g.f.: Product_{m>0} (1+x^m+x^(2*m)/2!).

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A293139 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} (-1)^j*x^(j*i)/j!.

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A293486 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*(2*i-1))/j!.

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A293299 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} 1/(Sum_{j=0..k} x^(j*i)/j!).

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A293301 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} 1/(Sum_{j=0..k} (-1)^j*x^(j*i)/j!).

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A293195 E.g.f.: Product_{m>0} (1+x^m+x^(2*m)/2!+x^(3*m)/3!).

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Links

Crossrefs

Programs

Maple

A293196 E.g.f.: Product_{m>0} (1+x^m+x^(2*m)/2!+x^(3*m)/3!+x^(4*m)/4!).

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Author

Keywords

Links

Crossrefs

A293197 E.g.f.: Product_{m>0} (1+x^m+x^(2*m)/2!+x^(3*m)/3!+x^(4*m)/4!+x^(5*m)/5!).

Views

Author

Keywords

Links

Crossrefs

Programs

Maple

A293139 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} (-1)^jx^(ji)/j!.

A293486 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(2i-1))/j!.

A293301 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} 1/(Sum_{j=0..k} (-1)^jx^(ji)/j!).

A293195 E.g.f.: Product_{m>0} (1+x^m+x^(2m)/2!+x^(3m)/3!).

A293196 E.g.f.: Product_{m>0} (1+x^m+x^(2m)/2!+x^(3m)/3!+x^(4*m)/4!).

A293197 E.g.f.: Product_{m>0} (1+x^m+x^(2m)/2!+x^(3m)/3!+x^(4m)/4!+x^(5m)/5!).