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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. 8
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 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,7

COMMENTS

The formula for the main diagonal, BesselI(0, 2*log(1 - x)), was found by Ilya Gutkovskiy (see A201950). - Paul D. Hanna, Feb 24 2019

LINKS

Paul D. Hanna, Table of n, a(n) for n = 0..2115

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).

EXAMPLE

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.

PROG

(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(""))}

CROSSREFS

Cf. A201950, A201951; diagonals: A201952, A201953.

Cf. A324305.

Sequence in context: A306245 A275043 A227061 * A291709 A326323 A257493

Adjacent sequences:  A201946 A201947 A201948 * A201950 A201951 A201952

KEYWORD

nonn,tabf

AUTHOR

Paul D. Hanna, Dec 06 2011

STATUS

approved

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Last modified October 22 22:22 EDT 2020. Contains 337962 sequences. (Running on oeis4.)