A366230 - OEIS

%I #14 Nov 19 2023 10:34:02

%S 1,1,0,2,2,0,6,18,3,0,24,144,96,4,0,120,1200,1800,400,5,0,720,10800,

%T 28800,16200,1440,6,0,5040,105840,441000,470400,119070,4704,7,0,40320,

%U 1128960,6773760,11760000,6021120,762048,14336,8,0,362880,13063680,106686720,274337280,238140000,65028096,4408992,41472,9,0

%N Expansion of e.g.f. A(x,y) satisfying A(x,y) = 1 + x*A(x,y) * exp(x*y * A(x,y)), as a triangle read by rows.

%C A161633(n) = Sum_{k=0..n} T(n,k) for n >= 0.

%C A366232(n) = Sum_{k=0..n} T(n,k) * 2^k for n >= 0.

%C A366233(n) = Sum_{k=0..n} T(n,k) * 3^k for n >= 0.

%C A366234(n) = Sum_{k=0..n} T(n,k) * 4^k for n >= 0.

%C A366235(n) = Sum_{k=0..n} T(n,k) * 5^k for n >= 0.

%F T(n,k) = n! * binomial(n+1, n-k)/(n+1) * (n-k)^k / k!.

%F Let A(x,y)^m = Sum_{n>=0} a(n,m) * x^n/n! then a(n,m) = n!*Sum_{k=0..n} binomial(n+m, n-k)*m/(n+m) * y^k * (n-k)^k/k!.

%F E.g.f. A(x,y) = Sum_{n>=0} x^n/n! * Sum_{k=0..n} T(n,k)*y^k satisfies the following formulas.

%F (1) A(x,y) = 1 + x*A(x) * exp(x*y*A(x,y)).

%F (2) A(x,y) = (1/x) * Series_Reversion( x/(1 + x*exp(x*y)) ).

%F (3) A( x/(1 + x*exp(x*y)), y) = 1 + x*exp(x*y).

%F (4) A(x,y) = 1 + (m+1) * Sum{n>=1} n*(n+m)^(n-2) * x^n/n! * A(x,y)^n * exp(-(n+m-y)*x*A(x,y)) for all fixed nonnegative m.

%F (4.a) A(x,y) = 1 + Sum{n>=1} n^(n-1) * x^n/n! * A(x,y)^n * exp(-(n-y)*x*A(x)).

%F (4.b) A(x,y) = 1 + 2 * Sum{n>=1} n*(n+1)^(n-2) * x^n/n! * A(x,y)^n * exp(-(n+1-y)*x*A(x,y)).

%F (4.c) A(x,y) = 1 + 3 * Sum{n>=1} n*(n+2)^(n-2) * x^n/n! * A(x,y)^n * exp(-(n+2-y)*x*A(x,y)).

%F (4.d) A(x,y) = 1 + 4 * Sum{n>=1} n*(n+3)^(n-2) * x^n/n! * A(x,y)^n * exp(-(n+3-y)*x*A(x,y)).

%F (4.e) A(x,y) = 1 + 5 * Sum{n>=1} n*(n+4)^(n-2) * x^n/n! * A(x,y)^n * exp(-(n+4-y)*x*A(x,y)).

%e E.g.f. A(x,y) = 1 + x + (2*y + 2)*x^2/2! + (3*y^2 + 18*y + 6)*x^3/3! + (4*y^3 + 96*y^2 + 144*y + 24)*x^4/4! + (5*y^4 + 400*y^3 + 1800*y^2 + 1200*y + 120)*x^5/5! + (6*y^5 + 1440*y^4 + 16200*y^3 + 28800*y^2 + 10800*y + 720)*x^6/6! + (7*y^6 + 4704*y^5 + 119070*y^4 + 470400*y^3 + 441000*y^2 + 105840*y + 5040)*x^7/7! + (8*y^7 + 14336*y^6 + 762048*y^5 + 6021120*y^4 + 11760000*y^3 + 6773760*y^2 + 1128960*y + 40320)*x^8/8! + ...

%e This triangle of coefficients T(n,k) of x^n*y^k/n! in A(x,y) begins

%e  1;

%e  1, 0;

%e  2, 2, 0;

%e  6, 18, 3, 0;

%e  24, 144, 96, 4, 0;

%e  120, 1200, 1800, 400, 5, 0;

%e  720, 10800, 28800, 16200, 1440, 6, 0;

%e  5040, 105840, 441000, 470400, 119070, 4704, 7, 0;

%e  40320, 1128960, 6773760, 11760000, 6021120, 762048, 14336, 8, 0;

%e  362880, 13063680, 106686720, 274337280, 238140000, 65028096, 4408992, 41472, 9, 0;

%e  ...

%o (PARI) {T(n,k) = n! * binomial(n+1, n-k)/(n+1) * (n-k)^k/k!}

%o for(n=0,10, for(k=0,n, print1(T(n,k),", "));print(""))

%Y Cf. A161633, A366232, A366233, A366234, A366235.

%K nonn,tabl

%O 0,4

%A _Paul D. Hanna_, Nov 17 2023