A245118 G.f. satisfies: A(x) = 1 + x*A(x)^3 / (A(x) - x*A'(x)).
1, 1, 3, 15, 103, 879, 8799, 99711, 1250175, 17077215, 251414175, 3958086015, 66245028447, 1173340901695, 21914481618591, 430320455540223, 8861927054203743, 190989549802914495, 4299479144808402975, 100927457271249951615, 2466702562122306371295, 62678175995432599506495
Offset: 0
Keywords
Examples
G.f.: A(x) = 1 + x + 3*x^2 + 15*x^3 + 103*x^4 + 879*x^5 + 8799*x^6 +... The table of coefficients of x^k in A(x)^n begin: n=1: [1, 1, 3, 15, 103, 879, 8799, 99711, 1250175, ...] n=2: [1, 2, 7, 36, 245, 2054, 20199, 225384, 2789545, ...] n=3: [1, 3, 12, 64, 435, 3597, 34806, 382590, 4674405, ...] n=4: [1, 4, 18, 100, 683, 5592, 53340, 577960, 6971325, ...] n=5: [1, 5, 25, 145, 1000, 8136, 76650, 819350, 9758835, ...] n=6: [1, 6, 33, 200, 1398, 11340, 105730, 1116024, 13129335, ...] n=7: [1, 7, 42, 266, 1890, 15330, 141736, 1478856, 17191251, ...] n=8: [1, 8, 52, 344, 2490, 20248, 186004, 1920552, 22071459, ...] n=9: [1, 9, 63, 435, 3213, 26253, 240069, 2455893, 27918000, ...] n=10:[1, 10, 75, 540, 4075, 33522, 305685, 3102000, 34903110, ...]; ... in which the diagonals illustrate the relation [x^n] A(x)^(n+1) = (n+1) * [x^(n-1)] A(x)^(n+2) for n>=1 as follows: [x^1] A(x)^2 = 2 = 2*[x^0] A(x)^3 = 2*1 ; [x^2] A(x)^3 = 12 = 3*[x^1] A(x)^4 = 3*4 ; [x^3] A(x)^4 = 100 = 4*[x^2] A(x)^5 = 4*25 ; [x^4] A(x)^5 = 1000 = 5*[x^3] A(x)^6 = 5*200 ; [x^5] A(x)^6 = 11340 = 6*[x^4] A(x)^7 = 6*1890 ; [x^6] A(x)^7 = 141736 = 7*[x^5] A(x)^8 = 7*20248 ; [x^7] A(x)^8 = 1920552 = 8*[x^6] A(x)^9 = 8*240069 ; [x^8] A(x)^9 = 27918000 = 9*[x^7] A(x)^10 = 9*3102000 ; ... RELATED SERIES. Also, from the above table we can generate A182304: [1/1, 2/2, 12/3, 100/4, 1000/5, 11340/6, 141736/7, 1920552/8, ...] = [1, 1, 4, 25, 200, 1890, 20248, 240069, 3102000, 43226590, ...]; the g.f. of A182304 begins: G(x) = x + x^2 + 4*x^3 + 25*x^4 + 200*x^5 + 1890*x^6 + 20248*x^7 +... where G(x) = x + d/dx G(x)^3/3 and G(x) = A(G(x)).
Links
- Vaclav Kotesovec, Table of n, a(n) for n = 0..300
Programs
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PARI
/* From [x^n] A(x)^(n+1) = (n+1) * [x^(n-1)] A(x)^(n+2): */ {a(n)=local(A=[1, 1]); for(i=2, n, A=concat(A, 0); A[#A]=((#A)*Vec(Ser(A)^(#A+1))[#A-1]-Vec(Ser(A)^(#A))[#A])/(#A)); A[n+1]} for(n=0,30,print1(a(n),", ")) -
PARI
/* From A(x) = 1 + x*A(x)^3 / (A(x) - x*A'(x)): */ {a(n)=local(A=1+x); for(i=1, n, A = 1 + x*A^3/(A - x*A' +x*O(x^n))); polcoeff(A,n)} for(n=0,30,print1(a(n),", "))
Formula
G.f. A(x) satisfies:
(1) [x^n] A(x)^(n+1) = (n+1) * [x^(n-1)] A(x)^(n+2) for n>=1.
(2) A(x) = x/Series_Reversion(G(x)) where G(x) = x + d/dx G(x)^3/3 is the g.f. of A182304; thus, G(x) = A(G(x)).
a(n) ~ c * n! * n^4, where c = 0.005428317993266202636748034138... (same as for A177384). - Vaclav Kotesovec, Jul 28 2014