A193111
G.f. satisfies: 1 = Sum_{n>=0} (-x)^(n*(n+1)/2) * A(x)^(n+1).
Original entry on oeis.org
1, 1, 2, 6, 19, 63, 218, 781, 2869, 10742, 40846, 157318, 612446, 2406100, 9527159, 37981611, 152328497, 614167702, 2487941464, 10121128882, 41330709103, 169362297620, 696187639438, 2870017515884, 11862845007114, 49152859179055
Offset: 0
G.f.: A(x) = 1 + x + 2*x^2 + 6*x^3 + 19*x^4 + 63*x^5 + 218*x^6 +...
which satisfies:
1 = A(x) - x*A(x)^2 - x^3*A(x)^3 + x^6*A(x)^4 + x^10*A(x)^5 - x^15*A(x)^6 - x^21*A(x)^7 ++--...
Related expansions.
A(x)^2 = 1 + 2*x + 5*x^2 + 16*x^3 + 54*x^4 + 188*x^5 + 674*x^6 +...
A(x)^3 = 1 + 3*x + 9*x^2 + 31*x^3 + 111*x^4 + 405*x^5 + 1505*x^6 +...
-
{a(n)=local(A=[1]); for(i=1, n, A=concat(A, 0); A[#A]=polcoeff(1-sum(m=0, sqrtint(2*(#A))+1, (-x)^(m*(m+1)/2)*Ser(A)^(m+1)), #A-1)); if(n<0, 0, A[n+1])}
A193112
G.f. satisfies: 1 = Sum_{n>=0} (-x)^(n*(n+1)/2) * A(x)^(2*n+1).
Original entry on oeis.org
1, 1, 3, 13, 63, 328, 1796, 10200, 59529, 354837, 2151079, 13221261, 82200739, 516053099, 3266812048, 20829635112, 133651716406, 862342656359, 5591505085491, 36416212224801, 238114435569354, 1562560513492974, 10287406857203911
Offset: 0
G.f.: A(x) = 1 + x + 3*x^2 + 13*x^3 + 63*x^4 + 328*x^5 + 1796*x^6 +...
which satisfies:
1 = A(x) - x*A(x)^3 - x^3*A(x)^5 + x^6*A(x)^7 + x^10*A(x)^9 - x^15*A(x)^11 - x^21*A(x)^13 ++--...
Related expansions.
A(x)^3 = 1 + 3*x + 12*x^2 + 58*x^3 + 303*x^4 + 1662*x^5 + 9447*x^6 +...
A(x)^5 = 1 + 5*x + 25*x^2 + 135*x^3 + 760*x^4 + 4401*x^5 +...
-
{a(n)=local(A=[1]); for(i=1, n, A=concat(A, 0); A[#A]=polcoeff(1-sum(m=0, sqrtint(2*(#A))+1, (-x)^(m*(m+1)/2)*Ser(A)^(2*m+1)), #A-1)); if(n<0, 0, A[n+1])}
A193113
G.f. satisfies: 1 = Sum_{n>=0} (-x)^(n*(n+1)/2) * A(x)^(3*n+1).
Original entry on oeis.org
1, 1, 4, 23, 151, 1074, 8059, 62814, 503619, 4126954, 34411602, 291025337, 2490377810, 21523367553, 187603609077, 1647252368595, 14556722879278, 129366008725176, 1155458240271571, 10366549508487178, 93382085749705066, 844255894224907354
Offset: 0
G.f.: A(x) = 1 + x + 4*x^2 + 23*x^3 + 151*x^4 + 1074*x^5 + 8059*x^6 +...
which satisfies:
1 = A(x) - x*A(x)^4 - x^3*A(x)^7 + x^6*A(x)^10 + x^10*A(x)^13 - x^15*A(x)^16 - x^21*A(x)^19 ++--...
Related expansions.
A(x)^4 = 1 + 4*x + 22*x^2 + 144*x^3 + 1025*x^4 + 7696*x^5 +...
A(x)^7 = 1 + 7*x + 49*x^2 + 364*x^3 + 2814*x^4 + 22400*x^5 +...
-
{a(n)=local(A=[1]); for(i=1, n, A=concat(A, 0); A[#A]=polcoeff(1-sum(m=0, sqrtint(2*(#A))+1, (-x)^(m*(m+1)/2)*Ser(A)^(3*m+1)), #A-1)); if(n<0, 0, A[n+1])}
A193115
G.f. satisfies: 1 = Sum_{n>=0} (-x)^(n^2) * A(x)^(2*n+1).
Original entry on oeis.org
1, 1, 3, 12, 54, 265, 1373, 7388, 40888, 231250, 1330618, 7764670, 45841323, 273316120, 1643345418, 9953021248, 60665811025, 371850104167, 2290623433302, 14173331572490, 88049709138896, 548978010516319, 3434070688405887, 21545961024510032
Offset: 0
G.f.: A(x) = 1 + x + 3*x^2 + 12*x^3 + 54*x^4 + 265*x^5 + 1373*x^6 +...
which satisfies:
1 = A(x) - x*A(x)^3 + x^4*A(x)^5 - x^9*A(x)^7 + x^16*A(x)^9 -+...
Related expansions.
A(x)^3 = 1 + 3*x + 12*x^2 + 55*x^3 + 270*x^4 + 1398*x^5 + 7518*x^6 +...
A(x)^5 = 1 + 5*x + 25*x^2 + 130*x^3 + 695*x^4 + 3816*x^5 +...
A193116
G.f. satisfies: 1 = Sum_{n>=0} (-x)^(n^2) * A(x)^(3*n+1).
Original entry on oeis.org
1, 1, 4, 22, 139, 958, 6979, 52851, 411884, 3281684, 26609931, 218874331, 1821767351, 15315464340, 129859965329, 1109239893974, 9536166375605, 82449167265098, 716449009997437, 6253709697731562, 54808237437608982, 482103739329417219
Offset: 0
G.f.: A(x) = 1 + x + 4*x^2 + 22*x^3 + 139*x^4 + 958*x^5 + 6979*x^6 +...
which satisfies:
1 = A(x) - x*A(x)^4 + x^4*A(x)^7 - x^9*A(x)^10 + x^16*A(x)^13 -+...
Related expansions.
A(x)^4 = 1 + 4*x + 22*x^2 + 140*x^3 + 965*x^4 + 7028*x^5 +...
A(x)^7 = 1 + 7*x + 49*x^2 + 357*x^3 + 2688*x^4 + 20811*x^5 +...
-
{a(n)=local(A=[1]); for(i=1, n, A=concat(A, 0); A[#A]=polcoeff(1-sum(m=0, sqrtint(#A)+1, (-x)^(m^2)*Ser(A)^(3*m+1) ), #A-1)); if(n<0, 0, A[n+1])}
Showing 1-5 of 5 results.