A152555 -id:A152555 - cp's OEIS frontend

A152559 Largest term in row n of triangle A152555.

1, 2, 7, 42, 495, 7436, 153361, 3853322, 116503839, 4127612326, 167061660005, 7671346786170, 392298901133895, 22113791358359574, 1365967717507556804, 91549507266620360316, 6624242349107514460269, 514871138228210665592112

Offset: 0

Paul D. Hanna, Dec 07 2008

nonn

Compare to row sums of triangle A152555: 2*(2n+2)^(n-1).

Triangle A152555 lists coefficients in a q-analog of the function LambertW(-2x)/(-2x).

Cf. A152555, A152556(q=-1), A152557 (q=2) A152558 (q=3).

{a(n)=local(e_q=1+sum(j=1,n,x^j/prod(i=1,j,(q^i-1)/(q-1))), LW2_q=serreverse(x/(e_q+x*O(x^n))^2)/x); vecsort(Vec(polcoeff(LW2_q+x*O(x^n),n,x)*prod(i=1,n,(q^i-1)/(q-1))))[n*(n-1)/2+1]}

A152550 Coefficients in a q-analog of the function [LambertW(-2x)/(-2x)]^(1/2), as a triangle read by rows.

Original entry on oeis.org

1, 1, 3, 2, 12, 16, 16, 5, 55, 110, 170, 180, 130, 70, 14, 273, 728, 1443, 2145, 2640, 2614, 2200, 1485, 783, 288, 42, 1428, 4760, 11312, 20657, 32032, 42833, 50477, 52934, 49441, 41069, 29876, 19019, 10010, 4158, 1155, 132, 7752, 31008, 85272

Offset: 0

Paul D. Hanna, Dec 07 2008

LambertW satisfies: [LambertW(-2x)/(-2x)]^(1/2) = exp(x*LambertW(-2x)/(-2x)).

			Triangle begins:
  1;
  1;
  3,2;
  12,16,16,5;
  55,110,170,180,130,70,14;
  273,728,1443,2145,2640,2614,2200,1485,783,288,42;
  1428,4760,11312,20657,32032,42833,50477,52934,49441,41069,29876,19019,10010,4158,1155,132;
  7752,31008,85272,181356,328440,521152,745416,969000,1159060,1278996,1307556,1238368,1085488,877240,650052,437164,262964,138320,60424,20592,4576,429;...
where row sums = (2*n+1)^(n-1) (A052750).
Row sums at q=-1 = (2*n+1)^[(n-1)/2] (A152551).
The generating function starts:
A(x,q) = 1 + x + (3 + 2*q)*x^2/faq(2,q) + (12 + 16*q + 16*q^2 + 5*q^3)*x^3/faq(3,q) + (55 + 110*q + 170*q^2 + 180*q^3 + 130*q^4 + 70*q^5 + 14*q^6)*x^4/faq(4,q) + ...
G.f. satisfies: A(x,q) = e_q( x*A(x,q)^2, q) where q-exponential series: e_q(x,q) = 1 + x + x^2/faq(2,q) + x^3/faq(3,q) +...+ x^n/faq(n,q) +...
The q-factorial of n is faq(n,q) = Product_{k=1..n} (q^k-1)/(q-1): faq(0,q)=1, faq(1,q)=1, faq(2,q)=(1+q), faq(3,q)=(1+q)*(1+q+q^2), faq(4,q)=(1+q)*(1+q+q^2)*(1+q+q^2+q^3),...
Special cases.
q=0: A(x,0) = 1 + x + 3*x^2 + 12*x^3 + 55*x^4 + 273*x^5 +... (A001764)
q=1: A(x,1) = 1 + x + 5/2*x^2 + 49/6*x^3 + 729/24*x^4 + 14641/120*x^5 +...
q=2: A(x,2) = 1 + x + 7/3*x^2 + 148/21*x^3 + 7611/315*x^4 + 872341/9765*x^5 +...
q=3: A(x,3) = 1 + x + 9/4*x^2 + 339/52*x^3 + 44521/2080*x^4 + 19059921/251680*x^5 +...

Paul D. Hanna, Rows 0 to 30 of the triangle, flattened.
Eric Weisstein's World of Mathematics, q-Exponential Function.
Eric Weisstein's World of Mathematics, q-Factorial.

Cf. A052750 (row sums), A001764 (column 0), A000108 (right border), A152554.
Cf. A152551 (q=-1), A152552 (q=2), A152553 (q=3).
Cf. variants: A152290, A152555.

{T(n,k)=local(e_q=1+sum(j=1,n,x^j/prod(i=1,j,(q^i-1)/(q-1))), LW2_q=sqrt(serreverse(x/(e_q+x*O(x^n))^2)/x)); polcoeff(polcoeff(LW2_q+x*O(x^n),n,x)*prod(i=1,n,(q^i-1)/(q-1))+q*O(q^k),k,q)}

G.f.: A(x,q) = Sum_{n>=0} Sum_{k=0..n*(n-1)/2} T(n,k)*q^k*x^n/faq(n,q), where faq(n,q) is the q-factorial of n.
G.f.: A(x,q) = [(1/x)*Series_Reversion( x/e_q(x,q)^2 )]^(1/2) where e_q(x,q) = Sum_{n>=0} x^n/faq(n,q) is the q-exponential function.
G.f. satisfies: A(x,q) = e_q( x*A(x,q)^2, q) and A( x/e_q(x,q)^2, q) = e_q(x,q).
G.f. at q=1: A(x,1) = (LambertW(-2*x)/(-2*x))^(1/2).
Row sums at q=+1: Sum_{k=0..n*(n-1)/2} T(n,k) = (2*n+1)^(n-1).
Row sums at q=-1: Sum_{k=0..n*(n-1)/2} T(n,k)*(-1)^k = (2*n+1)^[(n-1)/2].
Sum_{k=0..n*(n-1)/2} T(n,k)*exp(2*Pi*I*k/n) = 1 for n>=1; i.e., the n-th row sum at q = exp(2*Pi*I/n), the n-th root of unity, equals 1 for n>=1. - Vladeta Jovovic
Sum_{k=0..binomial(n,2)} T(n,k)*q^k = Sum_{pi} (2*n)!/(2*n-k+1)!*faq(n,q)/Product_{i=1..n} e(i)!*faq(i,q)^e(i), where pi runs through all nonnegative integer solutions of e(1)+2*e(2)+...+n*e(n) = n and k = e(1)+e(2)+...+e(n). - Vladeta Jovovic, Dec 04 2008

A152556 a(n) = 2(2n+2)^floor((n-1)/2).

Original entry on oeis.org

1, 2, 2, 16, 20, 288, 392, 8192, 11664, 320000, 468512, 15925248, 23762752, 963780608, 1458000000, 68719476736, 105046700288, 5642219814912, 8695584276992, 524288000000000, 813342767698944, 54394721876836352, 84841494965553152, 6232805962420322304

Offset: 0

Paul D. Hanna, Dec 07 2008

nonn

Compare to row sums of triangle A152555: 2*(2n+2)^(n-1).

Triangle A152555 lists coefficients in a q-analog of the function LambertW(-2x)/(-2x).

G. C. Greubel, Table of n, a(n) for n = 0..640

Cf. A152555, A152557(q=2), A152558 (q=3) A152559.

[2*(2*n+2)^(Floor((n-1)/2)): n in [0..30]]; // G. C. Greubel, Nov 17 2017

Table[2(2n+2)^Floor[(n-1)/2],{n,0,30}] (* Harvey P. Dale, Nov 13 2012 *)

PARI
```
a(n)=2*(2*n+2)^((n-1)\2)
```

a(n) = Sum_{k=0..n(n-1)/2} A152555(n,k)*(-1)^k.

A152557 Coefficients in a q-analog of the function LambertW(-2x)/(-2x) at q=2.

Original entry on oeis.org

1, 2, 17, 394, 21377, 2537724, 637139804, 332102399042, 355527029604321, 776504491956507890, 3445063827264105259985, 30955227335240072514768936, 562107762991597803740407081852, 20594660519301092842327319372549024

Offset: 0

Paul D. Hanna, Dec 07 2008

nonn

			G.f.: A(x) = 1 + 2*x + 17/3*x^2 + 394/21*x^3 + 21377/315*x^4 + 2537724/9765*x^5 +...
e_q(x,2) = 1 + x + x^2/3 + x^3/21 + x^4/315 + x^5/9765 + x^6/615195 +...
The q-factorial of n is faq(n,q) = Product_{k=1..n} (q^k-1)/(q-1).

Cf. A152555, A152556(q=-1), A152558 (q=3) A152559.

{a(n)=local(e_q=1+sum(j=1,n,x^j/prod(i=1,j,(q^i-1)/(q-1))), LW2_q=serreverse(x/(e_q+x*O(x^n))^2)/x); subst(polcoeff(LW2_q+x*O(x^n),n,x)*prod(i=1,n,(q^i-1)/(q-1)),q,2)}

G.f. satisfies: A(x) = e_q( x*A(x), 2)^2 and A( x/e_q(x,2)^2 ) = e_q(x,2)^2 where e_q(x,q) = Sum_{n>=0} x^n/faq(n,q) is the q-exponential function.
G.f.: A(x) = Sum_{n>=0} a(n)*x^n/faq(n,2) where faq(n,2) = q-factorial of n at q=2.
G.f.: A(x) = (1/x)*Series_Reversion( x/e_q(x,2)^2 )
a(n) = Sum_{k=0..n(n-1)/2} A152555(n,k)*2^k.

A152558 Coefficients in a q-analog of the function LambertW(-2x)/(-2x) at q=3.

Original entry on oeis.org

1, 2, 22, 912, 126692, 56277344, 78192313656, 335781903409152, 4424572027813470736, 178044609358672673825280, 21805611052892733414074516064, 8108006645142880473904973170212864

Offset: 0

Paul D. Hanna, Dec 07 2008

nonn

			G.f.: A(x) = 1 + 2*x + 22/4*x^2 + 912/52*x^3 + 126692/2080*x^4 + 56277344/251680*x^5 +...
e_q(x,3) = 1 + x + x^2/4 + x^3/52 + x^4/2080 + x^5/251680 + x^6/91611520 +...
The q-factorial of n is faq(n,q) = Product_{k=1..n} (q^k-1)/(q-1).

Cf. A152555, A152556(q=-1), A152557 (q=2) A152559.

{a(n)=local(e_q=1+sum(j=1,n,x^j/prod(i=1,j,(q^i-1)/(q-1))), LW2_q=serreverse(x/(e_q+x*O(x^n))^2)/x); subst(polcoeff(LW2_q+x*O(x^n),n,x)*prod(i=1,n,(q^i-1)/(q-1)),q,3)}

G.f. satisfies: A(x) = e_q( x*A(x), 3)^2 and A( x/e_q(x,3)^2 ) = e_q(x,3)^2 where e_q(x,q) = Sum_{n>=0} x^n/faq(n,q) is the q-exponential function.
G.f.: A(x) = Sum_{n>=0} a(n)*x^n/faq(n,3) where faq(n,3) = q-factorial of n at q=3.
G.f.: A(x) = (1/x)*Series_Reversion( x/e_q(x,3)^2 ).
a(n) = Sum_{k=0..n(n-1)/2} A152555(n,k)*3^k.

cp's OEIS Frontend

A152559 Largest term in row n of triangle A152555.

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A152550 Coefficients in a q-analog of the function [LambertW(-2x)/(-2x)]^(1/2), as a triangle read by rows.

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A152556 a(n) = 2*(2*n+2)^floor((n-1)/2).

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A152557 Coefficients in a q-analog of the function LambertW(-2x)/(-2x) at q=2.

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A152558 Coefficients in a q-analog of the function LambertW(-2x)/(-2x) at q=3.

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A152556 a(n) = 2(2n+2)^floor((n-1)/2).