A326001 G.f.: Sum_{n>=0} (n+1)*(n+2)/2 * ((1+x)^n - 1)^n.

1, 3, 24, 294, 4656, 89745, 2030628, 52649478, 1537164243, 49869371362, 1778978945148, 69186794933664, 2912819826915180, 131960308762527981, 6400097287173710814, 330837181757021507028, 18156728772490730152533, 1054301496336544498961490, 64575610215886231406950048, 4160633168635583272785917712, 281297401484498036336048099574, 19912063300892223213096919150032

Offset

0,2

Comments

More generally, the following sums are equal:

(1) Sum_{n>=0} binomial(n+k-1, n) * r^n * (p + q^n)^n,

(2) Sum_{n>=0} binomial(n+k-1, n) * r^n * q^(n^2) / (1 - p*q^n*r)^(n+k),

for any fixed integer k; here, k = 3 and p = -1, q = 1+x, r = 1.

Links

Table of n, a(n) for n=0..21.

Formula

Generating functions.

(1) Sum_{n>=0} (n+1)*(n+2)/2 * ((1+x)^n - 1)^n.

(2) Sum_{n>=0} (n+1)*(n+2)/2 * (1+x)^(n^2) / (1 + (1+x)^n)^(n+3).

Example

G.f.: A(x) = 1 + 3*x + 24*x^2 + 294*x^3 + 4656*x^4 + 89745*x^5 + 2030628*x^6 + 52649478*x^7 + 1537164243*x^8 + 49869371362*x^9 + 1778978945148*x^10 + ...
such that
A(x) = 1 + 3*((1+x)-1) + 6*((1+x)^2-1)^2 + 10*((1+x)^3-1)^3 + 15*((1+x)^4-1)^4 + 21*((1+x)^5-1)^5  + 28*((1+x)^6-1)^6 + 36*((1+x)^7-1)^7 + 45*((1+x)^8-1)^8 + 55*((1+x)^9-1)^9 +...
is equal to
A(x) = 1/2^3 + 3*(1+x)/(1+(1+x))^4 + 6*(1+x)^4/(1+(1+x)^2)^5 + 10*(1+x)^9/(1+(1+x)^3)^6 + 15*(1+x)^16/(1+(1+x)^4)^7 + 21*(1+x)^25/(1+(1+x)^5)^8 + 28*(1+x)^36/(1+(1+x)^6)^9 + 36*(1+x)^49/(1+(1+x)^7)^10 + ...

Prog

(PARI) {a(n) = my(A = sum(m=0, n, (m+1)*(m+2)/2 * ((1+x)^m - 1 +x*O(x^n))^m)); polcoeff(A, n)}
for(n=0, 25, print1(a(n), ", "))

Crossrefs

Cf. A122400, A326000.

Sequence in context: A059381 A301933 A337027 * A365998 A292186 A365567

Adjacent sequences: A325998 A325999 A326000 * A326002 A326003 A326004

Keyword

nonn

Author

Paul D. Hanna, Jun 01 2019

Status

approved