A201949 Triangle, read by rows, where the g.f. of row n equals Product_{k=0..n-1} (1 + k*y + y^2) for n>0 with a single '1' in row 0.
1, 1, 0, 1, 1, 1, 2, 1, 1, 1, 3, 5, 6, 5, 3, 1, 1, 6, 15, 24, 28, 24, 15, 6, 1, 1, 10, 40, 90, 139, 160, 139, 90, 40, 10, 1, 1, 15, 91, 300, 629, 945, 1078, 945, 629, 300, 91, 15, 1, 1, 21, 182, 861, 2520, 5019, 7377, 8358, 7377, 5019, 2520, 861, 182, 21, 1, 1, 28, 330, 2156, 8729, 23520, 45030, 65016, 73260, 65016
Offset: 0
Examples
E.g.f.: A(x,y) = 1 + (1 + y^2)*x + (1 + y + 2*y^2 + y^3 + y^4)*x^2/2! + (1 + 3*y + 5*y^2 + 6*y^3 + 5*y^4 + 3*y^5 + y^6)*x^3/3! + (1 + 6*y + 15*y^2 + 24*y^3 + 28*y^4 + 24*y^5 + 15*y^6 + 6*y^7 + y^8)*x^4/4! + (1 + 10*y + 40*y^2 + 90*y^3 + 139*y^4 + 160*y^5 + 139*y^6 + 90*y^7 + 40*y^8 + 10*y^9 + y^10)*x^5/5! + ...
which equals the power series expansion in x of the series given by
A(x,y) = Sum_{n>=0} log(1 - x*y)^(2*n) / (n!^2) - (1/y + y) * Sum_{n>=0} log(1 - x*y)^(2*n+1) / (n!*(n+1)!) + (1/y^2 + y^2) * Sum_{n>=0} log(1 - x*y)^(2*n+2) / (n!*(n+2)!) - (1/y3 + y^3) * Sum_{n>=0} (-log(1 - x*y))^(2*n+3) / (n!*(n+3)!) + (1/y^4 + y^4) * Sum_{n>=0} log(1 - x*y)^(2*n+4) / (n!*(n+4)!) + ...
Triangle begins:
[1],
[1, 0, 1],
[1, 1, 2, 1, 1],
[1, 3, 5, 6, 5, 3, 1],
[1, 6, 15, 24, 28, 24, 15, 6, 1],
[1, 10, 40, 90, 139, 160, 139, 90, 40, 10, 1],
[1, 15, 91, 300, 629, 945, 1078, 945, 629, 300, 91, 15, 1],
[1, 21, 182, 861, 2520, 5019, 7377, 8358, 7377, 5019, 2520, 861, 182, 21, 1],
[1, 28, 330, 2156, 8729, 23520, 45030, 65016, 73260, 65016, 45030, 23520, 8729, 2156, 330, 28, 1], ...
such that the g.f. of row n equals Product_{k=0..n-1} (1 + k*x + x^2) for n>0.
RELATED SERIES.
The e.g.f. may be defined by A(x,y) = x / Series_Reversion( F(x,y) )
where F(x,y) is the e.g.f. of triangle A324305 and equals
F(x,y) = Sum_{n>=1} x^n/n! * Product_{k=0..n-2} (n + k*y + n*y^2)
so that
F(x,y) = x + (2*y^2 + 2)*x^2/2! + (9*y^4 + 3*y^3 + 18*y^2 + 3*y + 9)*x^3/3! + (64*y^6 + 48*y^5 + 200*y^4 + 96*y^3 + 200*y^2 + 48*y + 64)*x^4/4! + (625*y^8 + 750*y^7 + 2775*y^6 + 2280*y^5 + 4300*y^4 + 2280*y^3 + 2775*y^2 + 750*y + 625)*x^5/5! + ...
where F(x,y) = Series_Reversion( x/A(x,y) ).
RELATED TRIANGLE.
Triangle A324305 of coefficients in F(x,y) such that F(x/A(x,y),y) = x begins
1;
2, 0, 2;
9, 3, 18, 3, 9;
64, 48, 200, 96, 200, 48, 64;
625, 750, 2775, 2280, 4300, 2280, 2775, 750, 625;
7776, 12960, 46440, 53640, 100584, 81360, 100584, 53640, 46440, 12960, 7776; ...
where the g.f. of row n is Product_{k=0..n-2} (n + k*y + n*y^2) for n >= 1.
Links
- Paul D. Hanna, Table of n, a(n) for n = 0..2115
Programs
-
PARI
{T(n,k)=polcoeff(prod(j=0,n-1,1+j*x+x^2),k)} {for(n=0,10,for(k=0,2*n,print1(T(n,k),","));print(""))}
Formula
Row sums yield the factorials.
Central terms in rows form A201950.
Antidiagonal sums yield A201951.
GENERATING FUNCTIONS.
E.g.f.: A(x,y) = 1/(1 - x*y)^(1/y + y). - Paul D. Hanna, Mar 02 2019
E.g.f.: A(x,y) = Sum_{k>=0} (1/y^k + y^k)/2^(0^k) * Sum_{n>=0} (-log(1 - x*y))^(2*n+k) / (n!*(n+k)!). - Paul D. Hanna, Feb 24 2019
E.g.f.: A(x,y) = x / Series_Reversion( F(x,y) ) such that F(x/A(x,y),y) = x, where F(x,y) = Sum_{n>=1} x^n/n! * Product_{k=0..n-2} (n + k*y + n*y^2) is the e.g.f. of A324305. - Paul D. Hanna, Feb 28 2019
E.g.f. of diagonal k: (1/y^k) * Sum_{n>=0} (-log(1 - x*y))^(2*n+k) / (n!*(n+k)!) for k >= 0. - Paul D. Hanna, Feb 24 2019
PARTICULAR ARGUMENTS.
E.g.f. at y = 0: A(x,y=0) = exp(x).
E.g.f. at y = 1: A(x,y=1) = 1/(1-x)^2.
E.g.f. at y = 2: A(x,y=2) = 1/(1-2*x)^(5/2).
Comments