A298689
G.f. A(x) satisfies: A(x) = Sum_{n>=0} binomial( n^2, n) * x^n / A(x)^( n^2 ).
Original entry on oeis.org
1, 1, 5, 56, 957, 22312, 666666, 24367474, 1051351629, 52144520972, 2915915251326, 181227240764128, 12382862552065170, 922234506009645794, 74345308066436693828, 6449466281781165675666, 599083515375854753327365, 59328642583049975996828036, 6240245388930730524658068558, 694754212357547941002786433000, 81628078642468462576697539116234
Offset: 0
G.f.: A(x) = 1 + x + 5*x^2 + 56*x^3 + 957*x^4 + 22312*x^5 + 666666*x^6 + 24367474*x^7 + 1051351629*x^8 + 52144520972*x^9 + 2915915251326*x^10 + 181227240764128*x^11 + 12382862552065170*x^12 + ...
such that
A(x) = 1 + C(1,1)*x/A(x) + C(4,2)*x^2/A(x)^4 + C(9,3)*x^3/A(x)^9 + C(16,4)*x^4/A(x)^16 + C(25,5)*x^5/A(x)^25 + C(36,6)*x^6/A(x)^36 + C(49,7)*x^7/A(x)^49 + ...
more explicitly,
A(x) = 1 + x/A(x) + 6*x^2/A(x)^4 + 84*x^3/A(x)^9 + 1820*x^4/A(x)^16 + 53130*x^5/A(x)^25 + 1947792*x^6/A(x)^36 + 85900584*x^7/A(x)^49 + ...
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{a(n) = my(A=[1]); for(i=1,n, A = Vec(sum(m=0,#A,binomial(m^2,m) * x^m/Ser(A)^(m^2) ))); A[n+1]}
for(n=0,30,print1(a(n),", "))
A298690
G.f. A(x) satisfies: A(x) = Sum_{n>=0} binomial( n*(n+1)/2, n) * x^n / A(x)^( n*(n+1)/2 ).
Original entry on oeis.org
1, 1, 2, 10, 83, 971, 14679, 271065, 5887674, 146573343, 4106195739, 127709962780, 4364136955874, 162503129082497, 6548680061635319, 283973223632787150, 13185195626147207058, 652695122347799336199, 34316223642036784123819, 1909798106976656110119169, 112165977515060359849066878, 6933265352057611483132200642
Offset: 0
G.f.: A(x) = 1 + x + 2*x^2 + 10*x^3 + 83*x^4 + 971*x^5 + 14679*x^6 + 271065*x^7 + 5887674*x^8 + 146573343*x^9 + 4106195739*x^10 + 127709962780*x^11 + 4364136955874*x^12 + 162503129082497*x^13 + 6548680061635319*x^14 + 283973223632787150*x^15 + ...
such that
A(x) = 1 + C(1,1)*x/A(x) + C(3,2)*x^2/A(x)^3 + C(6,3)*x^3/A(x)^6 + C(10,4)*x^4/A(x)^10 + C(15,5)*x^5/A(x)^15 + C(21,6)*x^6/A(x)^21 + C(28,7)*x^7/A(x)^28 + ...
more explicitly,
A(x) = 1 + x/A(x) + 3*x^2/A(x)^3 + 20*x^3/A(x)^6 + 210*x^4/A(x)^10 + 3003*x^5/A(x)^15 + 54264*x^6/A(x)^21 + 1184040*x^7/A(x)^28 + 30260340*x^8/A(x)^36 + ...
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{a(n) = my(A=[1]); for(i=1,n, A = Vec(sum(m=0,#A,binomial(m*(m+1)/2,m) * x^m/Ser(A)^(m*(m+1)/2) ))); G=Ser(A); A[n+1]}
for(n=0,30,print1(a(n),", "))
A298694
G.f. A(x) satisfies: A(x) = Sum_{n>=0} binomial( n*(n+1), n)/(n+1) * x^n / A(x)^(n^2).
Original entry on oeis.org
1, 1, 4, 32, 419, 8052, 207784, 6724274, 260396693, 11697865930, 596886780272, 34072732137625, 2151062784054901, 148819021611467291, 11198412956841549966, 910736443741061568539, 79616310026220269203631, 7446056807577515910468813, 741918566779386113373532994, 78467177619239380045368550016, 8779922184077661414128958823323
Offset: 0
G.f.: A(x) = 1 + x + 4*x^2 + 32*x^3 + 419*x^4 + 8052*x^5 + 207784*x^6 + 6724274*x^7 + 260396693*x^8 + 11697865930*x^9 + 596886780272*x^10 + 34072732137625*x^11 + 2151062784054901*x^12 + 148819021611467291*x^13 + 11198412956841549966*x^14 + 910736443741061568539*x^15 + ...
such that
A(x) = 1 + C(2,1)/2*x/A(x) + C(6,2)/3*x^2/A(x)^4 + C(12,3)/4*x^3/A(x)^9 + C(20,4)/5*x^4/A(x)^16 + C(30,5)/6*x^5/A(x)^25 + C(42,6)/7*x^6/A(x)^36 + C(56,7)/8*x^7/A(x)^49 + ...
more explicitly,
A(x) = 1 + x/A(x) + 5*x^2/A(x)^4 + 55*x^3/A(x)^9 + 969*x^4/A(x)^16 + 23751*x^5/A(x)^25 + 749398*x^6/A(x)^36 + 28989675*x^7/A(x)^49 + ... + A135861(n)*x^n/A(x)^(n^2) + ...
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terms = 21; A[] = 1; Do[A[x] = 1 + Sum[Binomial[n*(n+1), n]/(n+1)*x^n/ A[x]^(n^2), {n, terms}] + O[x]^terms, {terms}]; CoefficientList[A[x], x] (* Jean-François Alcover, Feb 09 2018 *)
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{a(n) = my(A=[1]); for(i=1, n, A = Vec(sum(m=0, #A, binomial(m*(m+1), m)/(m+1) * x^m/Ser(A)^(m^2) ))); A[n+1]}
for(n=0,20,print1(a(n),", "))
Showing 1-3 of 3 results.
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