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%I #23 Apr 12 2018 22:15:23
%S 1,1,4,16,86,544,3904,31328,276798,2660564,27576614,306051500,
%T 3615559236,45241980928,597141146374,8283583741588,120393776421550,
%U 1828261719906800,28937578248560784,476355010859517352,8139464481630136242,144109168217154747856,2639508261422244889106,49940898467864797567140,974790619672853340925800
%N G.f.: Sum_{n>=0} (1 + x*(1+x)^n)^n / 2^(n+1).
%C Row sums of triangle A300280.
%H Paul D. Hanna, <a href="/A300279/b300279.txt">Table of n, a(n) for n = 0..300</a>
%F G.f. is given by:
%F (1) Sum_{n>=0} (1 + x*(1+x)^n)^n / 2^(n+1).
%F (2) Sum_{n>=0} x^n * (1+x)^(n^2) / (2 - (1+x)^n)^(n+1).
%F Formulas for terms.
%F a(n) = Sum_{k=0..n} Sum_{j>=0} C(j+k, k) * C((j+k)*k, n-k) / 2^(j+k+1).
%e 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 + ...
%e such that
%e 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 + ...
%e Also, due to a series identity,
%e 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) + ...
%e Triangle A300280 is defined by
%e T(n,k) = Sum_{j>=0} C(j+k, k) * C((j+k)*k, n-k) / 2^(j+k+1), begins:
%e 1;
%e 0, 1;
%e 0, 3, 1;
%e 0, 5, 10, 1;
%e 0, 7, 57, 21, 1;
%e 0, 9, 252, 246, 36, 1;
%e 0, 11, 969, 2158, 710, 55, 1;
%e 0, 13, 3414, 15927, 10260, 1635, 78, 1; ...
%e the row sums of which form this sequence.
%e RELATED INFINITE SERIES.
%e At x = -1/2: the following sums are equal
%e S1 = Sum_{n>=1} (2^n - 1)^(n-1) / 2^(n^2),
%e S1 = Sum_{n>=1} (-1)^(n-1) / (2^n - 1)^n.
%e Explicitly,
%e 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 + ...
%e 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 + ...
%e where S1 = 0.891784622610953349715890136060239421022216970366139189336822360...
%o (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)}
%o for(n=0, 30, print1(a(n), ", "))
%Y Cf. A302765, A300280, A173217, A300050.
%K nonn
%O 0,3
%A _Paul D. Hanna_, Mar 01 2018
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