A157312 -id:A157312 - cp's OEIS frontend

A157311 G.f.: exp( Sum_{n>=1} a(n)x^n/n ) = Product_{n>=1} (1 + a(n-1)x^n).

1, 1, 1, 4, 13, 66, 394, 2759, 22005, 198049, 1979646, 21776107, 261287398, 3396736175, 47553219799, 713298307974, 11412712029909, 194016104508454, 3492285524896921, 66353424973041500, 1327068107226627278, 27868430252187313730, 613105422439139763585

Offset: 0

Paul D. Hanna, Mar 10 2009

nonn

			Define G(x) by the exponential:
G(x) = exp(x + x^2/2 + 4*x^3/3 + 13*x^4/4 + 66*x^5/5 + 394*x^6/6 +...)
then G(x) also equals the product:
G(x) = (1 + x)(1 + x^2)(1 + x^3)(1 + 4*x^4)(1 + 13*x^5)(1 + 66*x^6)*...;
where the coefficients in both expressions are the same (with offset)
and G(x) is the g.f. of A157312:
G(x) = 1 + x + x^2 + 2*x^3 + 5*x^4 + 18*x^5 + 84*x^6 + 481*x^7 + 3249*x^8 +...

Cf. A157312.

a[0] = 1; a[n_] := a[n] = DivisorSum[n, -# * (-a[#-1])^(n/#) &]; Array[a, 20, 0] (* Amiram Eldar, Aug 15 2023 *)

{a(n)=if(n==0,1,sumdiv(n,d,if(d>=1&d<=n,-d*(-a(d-1))^(n/d))))}

{a(n)=if(n==0, 1,n*polcoeff(1+sum(k=1,n,log(1+a(k-1)*x^k +x*O(x^n))),n))}

{a(n)=if(n==0, 1,n*polcoeff(sum(k=1,n,-sum(j=1,n\k,(-a(k-1))^j*x^(k*j)/j)+x*O(x^n)),n))}

a(n) = Sum_{d divides n, 1<=d<=n} -d*(-a(d-1))^(n/d) for n>0 with a(0)=1.

Product_{n>=1} (1 + a(n-1)*x^n) = g.f. of A157312.

A157314 G.f.: A(x) = exp( Sum_{n>=1} A157313(n)x^n/n ) = 1/Product_{n>=1} (1 - A157313(n-1)x^n).

Original entry on oeis.org

1, 1, 2, 5, 16, 62, 298, 1700, 11448, 88622, 778532, 7636888, 82782697, 981775224, 12643542295, 175638751080, 2617558335383, 41650633309937, 704712768652527, 12632584581030449, 239150363847113653, 4767657035201958150

Offset: 0

Paul D. Hanna, Mar 10 2009

nonn

			G.f.: A(x) = 1 + x + 2*x^2 + 5*x^3 + 16*x^4 + 62*x^5 + 298*x^6 +...
where the exponential:
A(x) = exp(x + 3*x^2/2 + 10*x^3/3 + 43*x^4/4 + 216*x^5/5 + 1326*x^6/6 +...)
and the product:
1/A(x) = (1 - x)(1 - x^2)(1 - 3*x^3)(1 - 10*x^4)(1 - 43*x^5)(1 - 216*x^6)*...
generate A(x) using the same coefficients (after initial term):
A157313=[1,1,3,10,43,216,1326,9283,74667,672085,6730098,74031079,...].

Cf. A157313, A157312.

cp's OEIS Frontend

A157311 G.f.: exp( Sum_{n>=1} a(n)x^n/n ) = Product_{n>=1} (1 + a(n-1)x^n).

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A157314 G.f.: A(x) = exp( Sum_{n>=1} A157313(n)x^n/n ) = 1/Product_{n>=1} (1 - A157313(n-1)x^n).

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cp's OEIS Frontend

A157311 G.f.: exp( Sum_{n>=1} a(n)*x^n/n ) = Product_{n>=1} (1 + a(n-1)*x^n).

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Author

Keywords

Examples

Crossrefs

Programs

Mathematica

PARI

PARI

PARI

Formula

A157314 G.f.: A(x) = exp( Sum_{n>=1} A157313(n)*x^n/n ) = 1/Product_{n>=1} (1 - A157313(n-1)*x^n).

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Author

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Examples

Crossrefs

A157311 G.f.: exp( Sum_{n>=1} a(n)x^n/n ) = Product_{n>=1} (1 + a(n-1)x^n).

A157314 G.f.: A(x) = exp( Sum_{n>=1} A157313(n)x^n/n ) = 1/Product_{n>=1} (1 - A157313(n-1)x^n).