A305109 O.g.f. A(x,y) satisfies: [x^n] exp( n * x*A(x,y) ) * (n + y - A(x,y)) = 0 for n > 0.

1, 0, 1, 2, 2, 2, 15, 30, 18, 6, 232, 492, 400, 144, 24, 5335, 12450, 11450, 5240, 1200, 120, 175416, 439698, 447744, 237612, 69672, 10800, 720, 7847665, 20851502, 22993348, 13653304, 4724328, 956760, 105840, 5040, 460083056, 1283257192, 1509767920, 979072928, 383250880, 92961216, 13700928, 1128960, 40320, 34295632587, 99690153120, 123801126966, 86244590412, 37181530008, 10307520792, 1842700968, 205630272, 13063680, 362880, 3171572232860, 9554655641770, 12424884790780, 9188854250660, 4284505197600, 1318457696960, 271837330560, 37189193040, 3241998720, 163296000, 3628800

Offset

0,4

Comments

Note: the factorial series, F(x) = Sum_{n>=0} n! * x^n, satisfies:

(1) [x^n] exp( n * x*F(x) ) * (2 - F(x)) = 0 for n > 0,

(2) [x^n] exp( x*F(x) ) * (n + 1 - F(x)) = 0 for n > 0.

It is remarkable that this triangle should consist entirely of integers.

Links

Paul D. Hanna, Table of n, a(n) for n = 0..1325 of rows 0..50 of this triangle in flattened form.

Example

G.f.: A(x,y) = 1 + x*y + x^2*(2 + 2*y + 2*y^2) + x^3*(15 + 30*y + 18*y^2 + 6*y^3) + x^4*(232 + 492*y + 400*y^2 + 144*y^3 + 24*y^4) + x^5*(5335 + 12450*y + 11450*y^2 + 5240*y^3 + 1200*y^4 + 120*y^5) + x^6*(175416 + 439698*y + 447744*y^2 + 237612*y^3 + 69672*y^4 + 10800*y^5 + 720*y^6) + x^7*(7847665 + 20851502*y + 22993348*y^2 + 13653304*y^3 + 4724328*y^4 + 956760*y^5 + 105840*y^6 + 5040*y^7) + x^8*(460083056 + 1283257192*y + 1509767920*y^2 + 979072928*y^3 + 383250880*y^4 + 92961216*y^5 + 13700928*y^6 + 1128960*y^7 + 40320*y^8) + ...
where A(x,y) = Sum_{n>=0} Sum_{k=0..n} T(n,k) * x^n*y^k satisfies:
[x^n] exp( n * x*A(x,y) ) * (n + y - A(x,y)) = 0 for n > 0.
This triangle of coefficients T(n,k) of x^n*y^k in A(x,y) begins:
1;
0, 1;
2, 2, 2;
15, 30, 18, 6;
232, 492, 400, 144, 24;
5335, 12450, 11450, 5240, 1200, 120;
175416, 439698, 447744, 237612, 69672, 10800, 720;
7847665, 20851502, 22993348, 13653304, 4724328, 956760, 105840, 5040;
460083056, 1283257192, 1509767920, 979072928, 383250880, 92961216, 13700928, 1128960, 40320;
34295632587, 99690153120, 123801126966, 86244590412, 37181530008, 10307520792, 1842700968, 205630272, 13063680, 362880; ...
in which the main diagonal equals the factorials.

Prog

(PARI) {T(n, k) = my(A=[1], m); for(i=1, n, A=concat(A, 0); m=#A; A[m] = Vec( exp( (m-1)*x*(Ser(A)) ) * (m-1 + y - Ser(A)) )[m] ); polcoeff(A[n+1], k)}
for(n=0, 10, for(k=0, n, print1(T(n, k), ", ")); print(""))

Crossrefs

Cf. A305110 (row sums), A305111 (column 0), A305112 (y=2), A305113 (y=3).

Sequence in context: A129409 A352029 A025521 * A372418 A068218 A098919

Adjacent sequences: A305106 A305107 A305108 * A305110 A305111 A305112

Keyword

nonn,tabl

Author

Paul D. Hanna, May 26 2018

Status

approved