A293731 -id:A293731 - cp's OEIS frontend

A215915 E.g.f.: exp( Sum_{n>=1} A000041(n)*x^n/n ), where A000041(n) is the number of partitions of n.

1, 1, 3, 13, 79, 579, 5209, 53347, 628257, 8223481, 119473291, 1893056781, 32677209103, 606930554923, 12109058077809, 257638964244739, 5830359141736129, 139638723615395697, 3531794326401241747, 93977250969358226701, 2625647922067519041231, 76809884197769914248211

Offset: 0

Paul D. Hanna, Aug 26 2012

nonn

Note that exp( Sum_{k>=1} A183610(n,k)*x^k/k ) is an integer series for row n>=1; the partition numbers, which forms row 0 of table A183610, is the exception.

			G.f.: A(x) = 1 + x + 3*x^2/2! + 13*x^3/3! + 79*x^4/4! + 579*x^5/5! + 5209*x^6/6! +  ...
such that log(A(x)) = x + 2*x^2/2 + 3*x^3/3 + 5*x^4/4 + 7*x^5/5 + 11*x^6/6 + 15*x^7/7 + 22*x^8/8 + ... + A000041(n)*x^n/n + ...

Seiichi Manyama, Table of n, a(n) for n = 0..432

Cf. A183610, A000041, A058892, A293731, A293839.

nmax = 20; CoefficientList[Series[E^Sum[PartitionsP[k]*x^k/k, {k, 1, nmax}], {x, 0, nmax}], x] * Range[0, nmax]! (* Vaclav Kotesovec, Oct 18 2017 *)

a(n):=if n=0 then 1 else (n-1)!*sum(num_partitions(i+1)*a(n-i-1)/(n-i-1)!,i,0,n-1); /* Vladimir Kruchinin, Feb 27 2015 */

{a(n)=n!*polcoeff(exp(sum(m=1,n+1,numbpart(m)*x^m/m+x*O(x^n))),n)}
for(n=0,31,print1(a(n),", "))

a(n) = (n-1)!*sum(p(i+1)*a(n-i-1)/(n-i-1)!,i,0,n-1), a(0)=1, where p(i) is the number of partitions of n. - Vladimir Kruchinin, Feb 27 2015

A293796 Square array A(n,k), n >= 0, k >= 0, read by antidiagonals, where column k is the expansion of e.g.f.: exp(Sum_{j>=1} j^(k-1)A000041(j)x^j).

Original entry on oeis.org

1, 1, 1, 1, 1, 3, 1, 1, 5, 13, 1, 1, 9, 31, 79, 1, 1, 17, 79, 265, 579, 1, 1, 33, 211, 937, 2621, 5209, 1, 1, 65, 583, 3433, 12501, 31621, 53347, 1, 1, 129, 1651, 12889, 62141, 204361, 426595, 628257, 1, 1, 257, 4759, 49225, 319461, 1395121, 3703099

Offset: 0

Seiichi Manyama, Oct 16 2017

			Square array begins:
     1,    1,     1,     1,      1, ...
     1,    1,     1,     1,      1, ...
     3,    5,     9,    17,     33, ...
    13,   31,    79,   211,    583, ...
    79,  265,   937,  3433,  12889, ...
   579, 2621, 12501, 62141, 319461, ...

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

Columns k=0..2 give A215915, A058892, A293731.

Rows n=0-1 give A000012.

A(0,k) = 1 and A(n,k) = (n-1)! * Sum_{j=1..n} j^k*A000041(j)*A(n-j,k)/(n-j)! for n > 0.

cp's OEIS Frontend

A215915 E.g.f.: exp( Sum_{n>=1} A000041(n)*x^n/n ), where A000041(n) is the number of partitions of n.

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A293796 Square array A(n,k), n >= 0, k >= 0, read by antidiagonals, where column k is the expansion of e.g.f.: exp(Sum_{j>=1} j^(k-1)A000041(j)x^j).

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cp's OEIS Frontend

A215915 E.g.f.: exp( Sum_{n>=1} A000041(n)*x^n/n ), where A000041(n) is the number of partitions of n.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

Maxima

PARI

Formula

A293796 Square array A(n,k), n >= 0, k >= 0, read by antidiagonals, where column k is the expansion of e.g.f.: exp(Sum_{j>=1} j^(k-1)*A000041(j)*x^j).

Views

Author

Keywords

Examples

Links

Crossrefs

Formula

A293796 Square array A(n,k), n >= 0, k >= 0, read by antidiagonals, where column k is the expansion of e.g.f.: exp(Sum_{j>=1} j^(k-1)A000041(j)x^j).