|
| |
|
|
A324958
|
|
Triangle of coefficients T(n,k) of y^n in Product_{k=0..n-2} (n + (2*n + k)*y + n*y^2), as read by rows of terms k = 0..2*n-2, for n >= 1.
|
|
3
|
|
|
|
1, 2, 4, 2, 9, 39, 60, 39, 9, 64, 432, 1160, 1584, 1160, 432, 64, 625, 5750, 22275, 47380, 60460, 47380, 22275, 5750, 625, 7776, 90720, 461160, 1343160, 2479464, 3029040, 2479464, 1343160, 461160, 90720, 7776, 117649, 1663893, 10489969, 38937360, 94679711, 158760987, 188149822, 158760987, 94679711, 38937360, 10489969, 1663893, 117649, 2097152, 34865152, 262635520, 1187049472, 3593318400, 7701010688, 12043471488, 13957194496, 12043471488, 7701010688, 3593318400, 1187049472, 262635520, 34865152, 2097152
(list;
graph;
refs;
listen;
history;
text;
internal format)
|
|
|
|
OFFSET
|
1,2
|
|
|
LINKS
|
|
|
|
FORMULA
|
E.g.f. A(x) = Sum_{n>=1} x^n/n! * Sum_{k=0..2*n-2} T(n,k)*y^k satisfies
(1) A(x,y) = Sum_{n>=1} x^n/n! * Product_{k=0..n-2} (n + (2*n + k)*y + n*y^2).
(2) A(x,y) = Series_Reversion( x*(1 - x*y)^((1+y)^2/y) ), wrt x.
(3) A(x,y) = x/(1 - y*A(x))^((1+y)^2/y).
(4) A(x,y) = x*Sum_{n>=0} A(x,y)^n/n! * Product_{k=0..n-1} (1 + (k+2)*y + y^2).
PARTICULAR ARGUMENTS.
E.g.f. at y = 0: A(x,y=0) = -LambertW(-x) = x*exp(-LambertW(-x)).
E.g.f. at y = 1: A(x,y=1) = x*G(x)^4, where G(x) = 1 + x*G(x)^5 is the g.f. of A002294.
FORMULAS INVOLVING TERMS.
Row sums: Sum_{k=0..2*n-2} T(n,k) = (5*n-2)!/(4*n-1)! for n >= 1.
T(n,0) = T(n,2*n-2) = n^(n-1) for n >= 1.
|
|
|
EXAMPLE
|
E.g.f.: A(x,y) = Sum_{n>=0} x^n/n! * Sum_{k=0..2*n-2} T(n,k)*y^k starts
A(x,y) = x + (2*y^2 + 4*y + 2)*x^2/2! + (9*y^4 + 39*y^3 + 60*y^2 + 39*y + 9)*x^3/3! + (64*y^6 + 432*y^5 + 1160*y^4 + 1584*y^3 + 1160*y^2 + 432*y + 64)*x^4/4! + (625*y^8 + 5750*y^7 + 22275*y^6 + 47380*y^5 + 60460*y^4 + 47380*y^3 + 22275*y^2 + 5750*y + 625)*x^5/5! + (7776*y^10 + 90720*y^9 + 461160*y^8 + 1343160*y^7 + 2479464*y^6 + 3029040*y^5 + 2479464*y^4 + 1343160*y^3 + 461160*y^2 + 90720*y + 7776)*x^6/6! + (117649*y^12 + 1663893*y^11 + 10489969*y^10 + 38937360*y^9 + 94679711*y^8 + 158760987*y^7 + 188149822*y^6 + 158760987*y^5 + 94679711*y^4 + 38937360*y^3 + 10489969*y^2 + 1663893*y + 117649)*x^7/7! + (2097152*y^14 + 34865152*y^13 + 262635520*y^12 + 1187049472*y^11 + 3593318400*y^10 + 7701010688*y^9 + 12043471488*y^8 + 13957194496*y^7 + 12043471488*y^6 + 7701010688*y^5 + 3593318400*y^4 + 1187049472*y^3 + 262635520*y^2 + 34865152*y + 2097152)*x^8/8! + ...
This triangle of coefficients T(n,k) of x^n*y^k/n! in e.g.f. A(x,y) begins:
1;
2, 4, 2;
9, 39, 60, 39, 9;
64, 432, 1160, 1584, 1160, 432, 64;
625, 5750, 22275, 47380, 60460, 47380, 22275, 5750, 625;
7776, 90720, 461160, 1343160, 2479464, 3029040, 2479464, 1343160, 461160, 90720, 7776;
117649, 1663893, 10489969, 38937360, 94679711, 158760987, 188149822, 158760987, 94679711, 38937360, 10489969, 1663893, 117649;
2097152, 34865152, 262635520, 1187049472, 3593318400, 7701010688, 12043471488, 13957194496, 12043471488, 7701010688, 3593318400, 1187049472, 262635520, 34865152, 2097152; ...
|
|
|
PROG
|
(PARI) {T(n, k) = polcoeff(prod(m=0, n-2, n + (2*n+m)*y + n*y^2 +y*O(y^k)), k, y)}
for(n=1, 10, for(k=0, 2*n-2, print1(T(n, k), ", ")); print(""))
(PARI) {T(n, k) = my(A = serreverse( x*(1 - x*y +x*O(x^n) )^((1+y)^2/y)));
n!*polcoeff(polcoeff(A, n, x), k, y)}
for(n=1, 10, for(k=0, 2*n-2, print1(T(n, k), ", ")); print(""))
|
|
|
CROSSREFS
|
|
|
|
KEYWORD
|
nonn,tabf
|
|
|
AUTHOR
|
|
|
|
STATUS
|
approved
|
| |
|
|
