A300279 G.f.: Sum_{n>=0} (1 + x*(1+x)^n)^n / 2^(n+1).

1, 1, 4, 16, 86, 544, 3904, 31328, 276798, 2660564, 27576614, 306051500, 3615559236, 45241980928, 597141146374, 8283583741588, 120393776421550, 1828261719906800, 28937578248560784, 476355010859517352, 8139464481630136242, 144109168217154747856, 2639508261422244889106, 49940898467864797567140, 974790619672853340925800

Offset

0,3

Comments

Row sums of triangle A300280.

Links

Paul D. Hanna, Table of n, a(n) for n = 0..300

Formula

G.f. is given by:

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

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

Formulas for terms.

a(n) = Sum_{k=0..n} Sum_{j>=0} C(j+k, k) * C((j+k)*k, n-k) / 2^(j+k+1).

Example

G.f.: A(x) = 1 + x + 4*x^2 + 16*x^3 + 86*x^4 + 544*x^5 + 3904*x^6 + 31328*x^7 + 276798*x^8 + 2660564*x^9 + 27576614*x^10 + ...
such that
A(x) = 1/2 + (1 + x*(1+x))/2^2 + (1 + x*(1+x)^2)^2/2^3 + (1 + x*(1+x)^3)^3/2^4 + (1 + x*(1+x)^4)^4/2^5 + (1 + x*(1+x)^5)^5/2^6 + (1 + x*(1+x)^6)^6/2^7 + ...
Also, due to a series identity,
A(x) = 1 + x*(1+x)/(2 - (1+x))^2 + x^2*(1+x)^4/(2 - (1+x)^2)^3 + x^3*(1+x)^9/(2 - (1+x)^3)^4 + x^4*(1+x)^16/(2 - (1+x)^4)^5 + x^5*(1+x)^25/(2 - (1+x)^5)^6 + x^6*(1+x)^36/(2 - (1+x)^6)^7 + ... + x^n * (1+x)^(n^2) / (2 - (1+x)^n)^(n+1) + ...
Triangle A300280 is defined by
T(n,k) = Sum_{j>=0} C(j+k, k) * C((j+k)*k, n-k) / 2^(j+k+1), begins:
1;
0, 1;
0, 3, 1;
0, 5, 10, 1;
0, 7, 57, 21, 1;
0, 9, 252, 246, 36, 1;
0, 11, 969, 2158, 710, 55, 1;
0, 13, 3414, 15927, 10260, 1635, 78, 1; ...
the row sums of which form this sequence.
RELATED INFINITE SERIES.
At x = -1/2: the following sums are equal
S1 = Sum_{n>=1} (2^n - 1)^(n-1) / 2^(n^2),
S1 = Sum_{n>=1} (-1)^(n-1) / (2^n - 1)^n.
Explicitly,
S1 = 1/2 + 3/2^4 + 7^2/2^9 + 15^3/2^16 + 31^4/2^25 + 63^5/2^36 + 127^6/2^49 + 255^7/2^64 + 511^8/2^81 + 1023^9/2^100 + 2047^10/2^121 + 4095^11/2^144 + ...
S1 = 1 - 1/3^2 + 1/7^3 - 1/15^4 + 1/31^5 - 1/63^6 + 1/127^7 - 1/255^8 + 1/511^9 - 1/1023^10 + 1/2047^11 - 1/4095^12 + 1/8191^13 - 1/16383^14 + ...
where S1 = 0.891784622610953349715890136060239421022216970366139189336822360...

Prog

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

Crossrefs

Cf. A302765, A300280, A173217, A300050.

Sequence in context: A238722 A184507 A165964 * A321238 A005618 A005495

Adjacent sequences: A300276 A300277 A300278 * A300280 A300281 A300282

Keyword

nonn

Author

Paul D. Hanna, Mar 01 2018

Status

approved