A161779 -id:A161779 - cp's OEIS frontend

A293266 Coefficients in asymptotic expansion of sequence A096161 and A161779.

1, 0, 1, 2, 7, 28, 121, 587, 3205, 19201, 123684, 850873, 6248839, 48948805, 407666212, 3594074850, 33405529547, 326310068618, 3342124657507, 35827145094057, 401325346421766, 4689964771177970, 57081316456694665, 722295766109273335, 9486188532177356598

Offset: 0

Vaclav Kotesovec, Oct 04 2017

nonn

			A096161(n) / n! ~ 1 + 1/n^2 + 2/n^3 + 7/n^4 + 28/n^5 + 121/n^6 + ...
A161779(n) / n! ~ 1 + 1/n^2 + 2/n^3 + 7/n^4 + 28/n^5 + 121/n^6 + ...

Vaclav Kotesovec, Table of n, a(n) for n = 0..122

Cf. A096161, A161779.

A293202 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} j!x^(ji).

Original entry on oeis.org

1, 1, 0, 1, 1, 0, 1, 1, 1, 0, 1, 1, 3, 2, 0, 1, 1, 3, 2, 2, 0, 1, 1, 3, 8, 6, 3, 0, 1, 1, 3, 8, 6, 7, 4, 0, 1, 1, 3, 8, 30, 13, 12, 5, 0, 1, 1, 3, 8, 30, 13, 24, 13, 6, 0, 1, 1, 3, 8, 30, 133, 48, 37, 22, 8, 0, 1, 1, 3, 8, 30, 133, 48, 61, 46, 26, 10, 0, 1, 1, 3, 8, 30, 133, 768, 181, 142, 98, 42, 12, 0

Offset: 0

Seiichi Manyama, Oct 02 2017

			Square array begins:
   1, 1, 1,  1,  1, ...
   0, 1, 1,  1,  1, ...
   0, 1, 3,  3,  3, ...
   0, 2, 2,  8,  8, ...
   0, 2, 6,  6, 30, ...
   0, 3, 7, 13, 13, ...

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

Columns k=0..5 give A000007, A000009, A293204, A289485, A289486, A293250.
Rows n=0..1 give A000012, A057427.
Main diagonal gives A161779.

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)-> 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_] := 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 *)

A293251 G.f.: Product_{i>0} 1/(Sum_{j>=0} j!x^(ji)).

Original entry on oeis.org

1, -1, -2, -3, -13, -65, -447, -3351, -28544, -269270, -2795872, -31689198, -389581679, -5165672203, -73512910689, -1117937393138, -18096787251877, -310743077434399, -5642249024063207, -108023468997424550, -2175086628366359447, -45952007357795606912

Offset: 0

Seiichi Manyama, Oct 04 2017

sign

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

Cf. A161779, A293259.

nmax = 30; CoefficientList[Series[Product[1/Sum[j!*x^(j*k), {j, 0, nmax}], {k, 1, nmax}], {x, 0, nmax}], x] (* Vaclav Kotesovec, Oct 04 2017 *)

Convolution inverse of A161779.

a(n) ~ -n! * (1 - 2/n - 2/n^2 - 5/n^3 - 29/n^4 - 232/n^5 - 2231/n^6 - 24745/n^7 - 308917/n^8 - 4279945/n^9 - 65179552/n^10), for coefficients see A293264. - Vaclav Kotesovec, Oct 04 2017

A333144 Irregular triangle where row n lists the product of the factorials of the exponentials of the partitions of n and the partitions are enumerated in canonical order.

Original entry on oeis.org

1, 1, 1, 2, 1, 1, 6, 1, 1, 2, 2, 24, 1, 1, 1, 2, 2, 6, 120, 1, 1, 1, 2, 2, 1, 6, 6, 4, 24, 720, 1, 1, 1, 2, 1, 1, 6, 2, 2, 2, 24, 6, 12, 120, 5040, 1, 1, 1, 2, 1, 1, 6, 2, 1, 2, 2, 24, 2, 4, 2, 6, 120, 24, 12, 48, 720, 40320

Offset: 0

Peter Luschny, Apr 10 2020

By 'canonical order' we understand the graded reverse lexicographic order (the default order of Mathematica and SageMath).

			The irregular table starts:
[0] [1]
[1] [1]
[2] [1, 2]
[3] [1, 1, 6]
[4] [1, 1, 2, 2, 24]
[5] [1, 1, 1, 2, 2, 6, 120]
[6] [1, 1, 1, 2, 2, 1, 6, 6, 4, 24, 720]
[7] [1, 1, 1, 2, 1, 1, 6, 2, 2, 2, 24, 6, 12, 120, 5040]

Row sums are A161779.

Cf. A069123.

def A333144row(n):
    return [product(factorial(expo) for expo in partition.to_exp()) for partition in Partitions(n)]
for n in (0..9): print(A333144row(n))

A161993 A006005 (shifted) convolved with all of its regularly "aerated" variants.

Original entry on oeis.org

1, 3, 8, 19, 43, 85, 171, 315, 580, 1022, 1766, 2982, 4959, 8081, 12997, 20596, 32261, 49909, 76447, 115872, 174133, 259312, 383206, 561877, 818225, 1183266, 1700658, 2429266, 3450562, 4874167, 6850072, 9578548, 13331445, 18469783, 25478494, 34999375, 47887091

Offset: 0

Gary W. Adamson, Jun 24 2009

nonn

Refer to A161779 for the analogous sequence based on the factorials.

Given A006005 (1 together with the odd primes = odd noncomposite numbers) = a, then b = the aerated variant: (1, 0, 3, 0, 5, 0, 7,...); c = (1, 0, 0, 3, 0, 0, 5,...) and so on such that A161993 = the infinite convolution product: a*b*c*...

Vaclav Kotesovec, Table of n, a(n) for n = 0..10000 (terms 0..5000 from Alois P. Heinz)

Cf. A006005, A161779.

p:= n-> `if`(n=0, 1, ithprime(n+1)):
b:= proc(n, i) option remember; `if`(i>n, 0,
     `if`(irem(n, i, 'r')=0, p(r), 0)+
      add(p(j)*b(n-i*j, i+1), j=0..n/i))
    end:
a:= n-> `if`(n=0, 1, b(n, 1)):
seq(a(n), n=0..45);  # Alois P. Heinz, Jul 27 2019

p[n_] := If[n==0, 1, Prime[n+1]];
b[n_, i_] := b[n, i] = If[i>n, 0, If[Mod[n, i]==0, p[n/i], 0] + Sum[p[j] b[n - i j, i+1], {j, 0, n/i}]];
a[n_] := If[n==0, 1, b[n, 1]];
a /@ Range[0, 45] (* Jean-François Alcover, Nov 20 2020, after Alois P. Heinz *)

Definition and comment corrected by Omar E. Pol, Aug 18 2011

Correct offset and a(13)-a(36) from Alois P. Heinz, Jul 27 2019

cp's OEIS Frontend

A293266 Coefficients in asymptotic expansion of sequence A096161 and A161779.

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A293202 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} j!*x^(j*i).

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A293251 G.f.: Product_{i>0} 1/(Sum_{j>=0} j!*x^(j*i)).

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Formula

A333144 Irregular triangle where row n lists the product of the factorials of the exponentials of the partitions of n and the partitions are enumerated in canonical order.

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Author

Keywords

Comments

Examples

Crossrefs

Programs

SageMath

A161993 A006005 (shifted) convolved with all of its regularly "aerated" variants.

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Author

Keywords

Comments

Links

Crossrefs

Programs

Maple

Mathematica

Extensions

A293202 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} j!x^(ji).

A293251 G.f.: Product_{i>0} 1/(Sum_{j>=0} j!x^(ji)).