login
The OEIS Foundation is supported by donations from users of the OEIS and by a grant from the Simons Foundation.

 

Logo


Hints
(Greetings from The On-Line Encyclopedia of Integer Sequences!)
A324305 Triangle, read by rows, where the g.f. of row n equals Product_{k=0..n-2} (n + k*y + n*y^2) for n > 1 with a single '1' in row 1. 4
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, 117649, 252105, 909979, 1337700, 2594501, 2753415, 3604342, 2753415, 2594501, 1337700, 909979, 252105, 117649, 2097152, 5505024, 20414464, 36040704, 73543680, 94730496, 133244544, 128389632, 133244544, 94730496, 73543680, 36040704, 20414464, 5505024, 2097152 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,2

LINKS

Paul D. Hanna, Table of n, a(n) for n = 1..10000 terms in rows 1..100 of this triangle in flattened form.

FORMULA

GENERATING FUNCTIONS.

E.g.f.: A(x,y) = x/(1 - y*A(x,y))^(1/y + y).

E.g.f.: A(x,y) = Series_Reversion( x*(1 - x*y)^(1/y + y) ), wrt x.

E.g.f.: A(x,y) = Sum_{n>=1} x^n/n! * Product_{k=0..n-2} (n + k*y + n*y^2)

E.g.f.: A(x,y) = Series_Reversion( x/G(x,y) ) such that A(x/G(x,y),y) = x, where G(x,y) = Sum_{n>=0} x^n/n! * Product_{k=0..n-1} (1 + k*y + y^2) is the e.g.f. of A201949.

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)^2, where G = 1 + x*G(x)^3 is the g.f. of A001764.

FORMULAS INVOLVING TERMS.

Row sums: Sum_{k=0..2*n-2} T(n,k) = (3*n-2)!/(2*n-1)! for n >= 1.

T(n,0) = T(n,2*n-2) = n^(n-1) for n >= 1.

T(n,n-1) = A324304(n) for n >= 1.

EXAMPLE

E.g.f.: A(x,y) = Sum_{n>=1} x^n/n! * Product_{k=0..n-2} (n + k*y + n*y^2) and satisfies A(x,y) = x/(1 - y*A(x,y))^(1/y + y).

Explicitly,

A(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! + (7776*y^10 + 12960*y^9 + 46440*y^8 + 53640*y^7 + 100584*y^6 + 81360*y^5 + 100584*y^4 + 53640*y^3 + 46440*y^2 + 12960*y + 7776)*x^6/6! + (117649*y^12 + 252105*y^11 + 909979*y^10 + 1337700*y^9 + 2594501*y^8 + 2753415*y^7 + 3604342*y^6 + 2753415*y^5 + 2594501*y^4 + 1337700*y^3 + 909979*y^2 + 252105*y + 117649)*x^7/7! + ...

Setting y = 1 yields an o.g.f. of A006013:

A(x,y=1) = x + 2*x^2 + 7*x^3 + 30*x^4 + 143*x^5 + 728*x^6 + 3876*x^7 + 21318*x^8 + 120175*x^9 + ... + binomial(3*n-2,n-1)/n * x^n + ...

TRIANGLE.

This triangle of coefficients in Product_{k=0..n-2} (n + k*y + n*y^2), n >= 1, 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;

117649, 252105, 909979, 1337700, 2594501, 2753415, 3604342, 2753415, 2594501, 1337700, 909979, 252105, 117649;

2097152, 5505024, 20414464, 36040704, 73543680, 94730496, 133244544, 128389632, 133244544, 94730496, 73543680, 36040704, 20414464, 5505024, 2097152; ...

RELATED SERIES.

The e.g.f. may be defined by A(x,y) = Series_Reversion( x/G(x,y) )

where G(x,y) is the e.g.f. of A201949 and equals

G(x,y) = Sum_{n>=0} x^n/n! * Product_{k=0..n-1} (1 + k*y + y^2)

so that

G(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! + ...

and G(x,y) = x / Series_Reversion( A(x,y) ).

RELATED TRIANGLE.

Triangle A201949 of coefficients in G(x,y) such that A(x/G(x,y),y) = x 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; ...

where the g.f. of row n is Product_{k=0..n-1} (1 + k*y + y^2) for n >= 0.

PROG

(PARI) {T(n, k)=polcoeff(prod(j=0, n-2,  n + j*y + n*y^2), k, y)}

{for(n=1, 10, for(k=0, 2*n-2, print1(T(n, k), ", ")); print(""))}

(PARI) /* A(x, y) = Series_Reversion(x/G(x, y)) where G(x, y) = e.g.f. A201949 */

{T(n, k) = my(G=1, A=x);

G = sum(m=0, n, x^m/m! * prod(j=0, m-1, 1 + j*y + y^2) +x*O(x^n));

A = serreverse(x/G);

n!*polcoeff(polcoeff(A, n, x), k, y)}

{for(n=1, 10, for(k=0, 2*n-2, print1(T(n, k), ", ")); print(""))}

CROSSREFS

Cf. A201949, A324304.

Cf. A324956, A324958.

Sequence in context: A344765 A259356 A137302 * A265607 A332628 A091518

Adjacent sequences:  A324302 A324303 A324304 * A324306 A324307 A324308

KEYWORD

nonn,tabf

AUTHOR

Paul D. Hanna, Feb 28 2019

STATUS

approved

Lookup | Welcome | Wiki | Register | Music | Plot 2 | Demos | Index | Browse | More | WebCam
Contribute new seq. or comment | Format | Style Sheet | Transforms | Superseeker | Recent
The OEIS Community | Maintained by The OEIS Foundation Inc.

License Agreements, Terms of Use, Privacy Policy. .

Last modified October 26 04:54 EDT 2021. Contains 348256 sequences. (Running on oeis4.)