A052829 -id:A052829 - cp's OEIS frontend

A052870 First differences of A052829.

1, 1, 2, 6, 15, 44, 128, 386, 1179, 3679, 11601, 37030, 119262, 387325, 1266647, 4168264, 13791565, 45856091, 153134306, 513403575, 1727395042, 5830866601, 19740613869, 67014421326, 228066659185, 777961702283, 2659398743509, 9109015516017, 31258117755635

Offset: 0

encyclopedia(AT)pommard.inria.fr, Jan 25 2000

Old name was: A simple grammar.

INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 841

Cf. A052829, A052855.

spec := [S,{C=Sequence(Z,1 <= card),S=PowerSet(B),B=Prod(C,S)},unlabeled]: seq(combstruct[count](spec,size=n), n=0..20);

From Seiichi Manyama, Jun 07 2023: (Start)
Conjectures: G.f. satisfies A(x) = exp( Sum_{k>=1} (-1)^(k+1) * A(x^k) * x^k/(k * (1 - x^k)) ).
A(x) = Sum_{k>=0} a(k) * x^k = Product_{j>=1} Product_{k>=0} (1+x^(j+k))^a(k). (End)

More terms from Alois P. Heinz, Mar 16 2016

A308227 G.f.: (x/(1 - x)) * Product_{k>=1} ((1 + x^k)/(1 - x^k))^a(k).

Original entry on oeis.org

1, 3, 11, 47, 217, 1065, 5453, 28789, 155633, 857207, 4793103, 27136555, 155249971, 896133487, 5212477023, 30522169103, 179777122393, 1064411910393, 6331361864657, 37817265028841, 226731778956181, 1363993567341257, 8231111557650837, 49812263080757845

Offset: 1

Ilya Gutkovskiy, May 16 2019

nonn

Cf. A029856, A036249, A052829, A073075.

a[n_] := a[n] = SeriesCoefficient[x/(1 - x) Product[((1 + x^k)/(1 - x^k))^a[k], {k, 1, n - 1}], {x, 0, n}]; Table[a[n], {n, 1, 24}]
terms = 24; A[] = 0; Do[A[x] = x Exp[Sum[2 A[x^(2 k - 1)]/(2 k - 1) + x^k/k, {k, 1, terms}]] + O[x]^(terms + 1) // Normal, terms + 1]; CoefficientList[A[x], x] // Rest

G.f. A(x) satisfies: A(x) = x * exp(Sum_{k>=1} 2*A(x^(2*k-1))/(2*k - 1) + x^k/k).

cp's OEIS Frontend

A052870 First differences of A052829.

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