A301309
G.f.: Sum_{n>=0} ( (1+x)^n + (1-x)^n )^n / 2^(2*n+1), an even function.
Original entry on oeis.org
1, 5, 418, 97248, 44494788, 33701146040, 38158722166012, 60370440881763184, 127193089522406873576, 344265367844128036044688, 1164086577885251318385747568, 4808913945776510766505317067088, 23831677319262549731059823149874928, 139543211306816620890086979219586374480, 953076439362156646686630002626476525309552
Offset: 0
G.f.: A(x) = 1 + 5*x^2 + 418*x^4 + 97248*x^6 + 44494788*x^8 + 33701146040*x^10 + 38158722166012*x^12 + 60370440881763184*x^14 + ...
such that
A(x) = 1/2 + ((1+x) + (1-x))/2^3 + ((1+x)^2 + (1-x)^2)^2/2^5 + ((1+x)^3 + (1-x)^3)^3/2^7 + ((1+x)^4 + (1-x)^4)^4/2^9 + ((1+x)^5 + (1-x)^5)^5/2^11 + ...
Equivalently,
A(x) = 1/2 + 1/2^2 + (1 + x^2)^2/2^3 + (1 + 3*x^2)^3/2^4 + (1 + 6*x^2 + x^4)^4/2^5 + (1 + 10*x^2 + 5*x^4)^5/2^6 + (1 + 15*x^2 + 15*x^4 + x^6)^6/2^7 + ...
A301465
G.f.: Sum_{n>=0} ( (1+x)^n + (1+2*x)^n )^n * x^n.
Original entry on oeis.org
1, 2, 7, 32, 180, 1142, 8162, 63807, 541202, 4926969, 47799429, 491254999, 5322191407, 60533874755, 720300053902, 8939841040157, 115427887156492, 1546880293065700, 21472465304481214, 308173758866549496, 4565492159926897329, 69713352146246150541, 1095715822838490323551, 17705238547423980566108, 293792275476690070452598
Offset: 0
G.f.: A(x) = 1 + 2*x + 7*x^2 + 32*x^3 + 180*x^4 + 1142*x^5 + 8162*x^6 + 63807*x^7 + 541202*x^8 + 4926969*x^9 + 47799429*x^10 + ...
such that
A(x) = 1 + ((1+x) + (1+2*x))*x + ((1+x)^2 + (1+2*x)^2)^2*x^2 + ((1+x)^3 + (1+2*x)^3)^3*x^3 + ((1+x)^4 + (1+2*x)^4)^4*x^4 + ((1+x)^5 + (1+2*x)^5)^5*x^5 + ...
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{a(n) = my(A=1); A = sum(k=0, n, ((1+x)^k + (1+2*x)^k +x*O(x^n))^k * x^k ); polcoeff(A, n)}
for(n=0, 30, print1(a(n), ", "))
A302108
G.f.: Sum_{n>=0} ( (1+x)^n - (1-x)^n )^n / 2^n.
Original entry on oeis.org
1, 1, 4, 27, 256, 3152, 47680, 854802, 17711872, 416372620, 10947581056, 318304921165, 10140097538560, 351219420860694, 13141237470041536, 528208859187285899, 22698715714385041920, 1038485165851106374784, 50395972495225521776384, 2585595617532863164095240, 139835798146777767415142912, 7950987913261988583226011167
Offset: 0
G.f.: A(x) = 1 + x + 4*x^2 + 27*x^3 + 256*x^4 + 3152*x^5 + 47680*x^6 + 854802*x^7 + 17711872*x^8 + 416372620*x^9 + 10947581056*x^10 + ...
such that
A(x) = 1 + ((1+x) - (1-x))/2 + ((1+x)^2 - (1-x)^2)^2/2^2 + ((1+x)^3 - (1-x)^3)^3/2^3 + ((1+x)^4 - (1-x)^4)^4/2^4 + ((1+x)^5 - (1-x)^5)^5/2^5 + ((1+x)^6 - (1-x)^6)^6/2^6 + ((1+x)^7 - (1-x)^7)^7/2^7 + ...
Equivalently,
A(x) = 1 + x + (2*x)^2 + (3*x + x^3)^3 + (4*x + 4*x^3)^4 + (5*x + 10*x^3 + x^5)^5 + (6*x + 20*x^3 + 6*x^5)^6 + (7*x + 35*x^3 + 21*x^5 + x^7)^7 + (8*x + 56*x^3 + 56*x^5 + 8*x^7)^8 + (9*x + 84*x^3 + 126*x^5 + 36*x^7 + x^9)^9 + (10*x + 120*x^3 + 252*x^5 + 120*x^7 + 10*x^9)^10 + ...
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{a(n) = my(A=1); A = sum(m=0, n, ((1+x)^m - (1-x)^m +x*O(x^n))^m/2^m ); polcoeff(A, n)}
for(n=0, 30, print1(a(n), ", "))
Showing 1-3 of 3 results.
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