A245464 G.f.: Sum_{n>=0} x^n * Sum_{k=0..n} C(n,k) * ((1+x)^k - 1)^k.

1, 1, 2, 3, 8, 21, 70, 263, 1072, 4842, 23351, 120478, 660372, 3817413, 23213642, 147866712, 983535760, 6814069842, 49050260795, 366092901787, 2827792333274, 22566873540299, 185782024439055, 1575592459104692, 13748110774214480, 123281851161743801, 1134880801686963605

Offset

0,3

Links

Paul D. Hanna, Table of n, a(n) for n = 0..200

Formula

G.f.: Sum_{n>=0} x^n * Sum_{k=0..n} C(n,k) * (1 - (1+x)^k)^(n-k) * (1+x)^(k^2).

G.f.: Sum_{n>=0} (1+x)^(n^2) * x^n / (1-x + x*(1+x)^n)^(n+1). - Paul D. Hanna, Jan 20 2015

Example

G.f.: A(x) = 1 + x + 2*x^2 + 3*x^3 + 8*x^4 + 21*x^5 + 70*x^6 + 263*x^7 +...
where we have the identity:
A(x) = 1 + x*(1 + ((1+x)-1))
+ x^2*(1 + 2*((1+x)-1) + ((1+x)^2-1)^2)
+ x^3*(1 + 3*((1+x)-1) + 3*((1+x)^2-1)^2 + ((1+x)^3-1)^3)
+ x^4*(1 + 4*((1+x)-1) + 6*((1+x)^2-1)^2 + 4*((1+x)^3-1)^3 + ((1+x)^4-1)^4)
+ x^5*(1 + 5*((1+x)-1) + 10*((1+x)^2-1)^2 + 10*((1+x)^3-1)^3 + 5*((1+x)^4-1)^4 + ((1+x)^5-1)^5) +...
is equal to
A(x) = 1 + x*(0 + (1+x))
+ x^2*(0 + 2*(1-(1+x))*(1+x) + (1+x)^4)
+ x^3*(0 + 3*(1-(1+x))^2*(1+x) + 3*(1-(1+x)^2)*(1+x)^4 + (1+x)^9)
+ x^4*(0 + 4*(1-(1+x))^3*(1+x) + 6*(1-(1+x)^2)^2*(1+x)^4 + 4*(1-(1+x)^3)*(1+x)^9 + (1+x)^16)
+ x^5*(0 + 5*(1-(1+x))^4*(1+x) + 6*(1-(1+x)^2)^3*(1+x)^4 + 4*(1-(1+x)^3)^2*(1+x)^9 + 5*(1-(1+x)^4)*(1+x)^16 + (1+x)^25) +...
Also,
A(x) = 1 + (1+x)*x/(1-x + x*(1+x))^2 + (1+x)^4*x^2/(1-x + x*(1+x)^2)^3 + (1+x)^9*x^3/(1-x + x*(1+x)^3)^4 + (1+x)^16*x^4/(1-x + x*(1+x)^4)^5 + (1+x)^25*x^5/(1-x + x*(1+x)^5)^6 + (1+x)^36*x^6/(1-x + x*(1+x)^6)^7 +...

Prog

(PARI) {a(n) = polcoeff(sum(m=0, n, x^m*sum(k=0, m, binomial(m, k) * ((1+x)^k - 1 +x*O(x^n))^k )) , n)}
for(n=0, 30, print1(a(n), ", "))
(PARI) {a(n) = polcoeff(sum(m=0, n, x^m*sum(k=0, m, binomial(m, k) * (1 - (1+x)^k +x*O(x^n))^(m-k) * (1+x)^(k^2) )) , n)}
for(n=0, 30, print1(a(n), ", "))
(PARI) {a(n)=local(A=1); A=sum(m=0, n, (1+x)^(m^2) * x^m / (1-x + x*(1+x)^m +x*O(x^n))^(m+1) ); polcoeff(A, n)}
for(n=0, 30, print1(a(n), ", "))

Crossrefs

Cf. A245465.

Sequence in context: A122263 A132730 A004790 * A243562 A238329 A272333

Adjacent sequences: A245461 A245462 A245463 * A245465 A245466 A245467

Keyword

nonn

Author

Paul D. Hanna, Jul 22 2014

Status

approved