A152550 -id:A152550 - cp's OEIS frontend

A152554 Largest term in row n of triangle A152550.

1, 1, 3, 16, 180, 2640, 52934, 1307556, 39067428, 1369499060, 54995284784, 2507211396061, 127388480252917, 7144814127814222, 439553511977812220, 29347225935730588372, 2116793087420823777580, 164035715631344596393196

Offset: 0

Paul D. Hanna, Dec 07 2008

nonn

Compare to row sums of triangle A152550: (2n+1)^(n-1).

Triangle A152550 lists coefficients in a q-analog of the function [LambertW(-2x)/(-2x)]^(1/2).

Cf. A152550, A152551 (q=-1), A152552 (q=2), A152553 (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=sqrt(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]}

A152555 Coefficients in a q-analog of the function LambertW(-2x)/(-2x), as a triangle read by rows.

Original entry on oeis.org

1, 2, 7, 5, 30, 42, 42, 14, 143, 297, 462, 495, 363, 198, 42, 728, 2002, 4004, 6006, 7436, 7436, 6292, 4290, 2288, 858, 132, 3876, 13260, 31824, 58604, 91364, 122876, 145535, 153361, 143936, 120185, 87971, 56329, 29939, 12584, 3575, 429, 21318, 87210

Offset: 0

Paul D. Hanna, Dec 07 2008

			Triangle begins:
  1;
  2;
  7,5;
  30,42,42,14;
  143,297,462,495,363,198,42;
  728,2002,4004,6006,7436,7436,6292,4290,2288,858,132;
  3876,13260,31824,58604,91364,122876,145535,153361,143936,120185,87971,56329,29939,12584,3575,429;
  21318,87210,242250,519384,945744,1508070,2165664,2826420,3392520,3756626,3853322,3662106,3221330,2613240,1944324,1313760,794614,420784,185640,64090,14586,1430;...
where row sums = 2*(2*n+2)^(n-1) (A097629).
Row sums at q=-1 = 2*(2*n+2)^[(n-1)/2] (A152556).
The generating function starts:
A(x,q) = 1 + 2*x + (7 + 5*q)*x^2/faq(2,q) + (30 + 42*q + 42*q^2 + 14*q^3)*x^3/faq(3,q) + (143 + 297*q + 462*q^2 + 495*q^3 + 363*q^4 + 198*q^5 + 42*q^6)*x^4/faq(4,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 + 2*x + 7*x^2 + 30*x^3 + 143*x^4 + 728*x^5 +... (A006013)
q=1: A(x,1) = 1 + 2*x + 12/2*x^2 + 128/6*x^3 + 2000/24*x^4 + 41472/120*x^5 +...
q=2: A(x,2) = 1 + 2*x + 17/3*x^2 + 394/21*x^3 + 21377/315*x^4 + 2537724/9765*x^5 +...
q=3: A(x,3) = 1 + 2*x + 22/4*x^2 + 912/52*x^3 + 126692/2080*x^4 + 56277344/251680*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. A097629 (row sums), A006013 (column 0), A000108 (right border), A152559.
Cf. A152556 (q=-1), A152557 (q=2), A152558 (q=3).
Cf. variants: A152290, A152550.

{T(n,k)=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); 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 ) 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), q)^2 and A( x/e_q(x,q)^2, q) = e_q(x,q)^2.
G.f. at q=1: A(x,1) = LambertW(-2*x)/(-2*x).
Row sums at q=+1: Sum_{k=0..n*(n-1)/2} T(n,k) = 2*(2*n+2)^(n-1).
Row sums at q=-1: Sum_{k=0..n*(n-1)/2} T(n,k)*(-1)^k = 2*(2*n+2)^[(n-1)/2].
Sum_{k=0..n*(n-1)/2} T(n,k)*exp(2*Pi*I*k/n) = 2 for n>=1; i.e., the n-th row sum at q = exp(2*Pi*I/n), the n-th root of unity, equals 2 for n>=1. - Vladeta Jovovic
Sum_{k=0..binomial(n,2)} T(n,k)*q^k = Sum_{pi} 2*(2*n+1)!/(2*n-k+2)!*faq(n,q)/Product_{i=1..n} e(i)!*faq(i,q)^e(i), where pi runs over all nonnegative integer solutions to e(1)+2*e(2)+...+n*e(n) = n and k = e(1)+e(2)+...+e(n). - Vladeta Jovovic, Dec 07 2008

A152800 Irregular triangle read by rows: the q-analog of the Euler numbers; expansion of the arithmetic inverse of the q-cosine of x.

Original entry on oeis.org

1, 1, 0, 1, 2, 1, 1, 0, 0, 1, 3, 5, 8, 10, 10, 9, 7, 5, 2, 1, 0, 0, 0, 1, 4, 10, 21, 36, 55, 78, 101, 122, 138, 145, 143, 134, 117, 95, 72, 50, 32, 18, 9, 3, 1, 0, 0, 0, 0, 1, 5, 16, 41, 87, 164, 283, 452, 679, 967, 1311, 1700, 2118, 2540, 2937, 3282, 3546, 3706, 3751, 3676, 3487

Offset: 0

Paul D. Hanna, Dec 26 2008

The q-cosine is cos_q(x,q) = Sum_{n>=0} (-1)^n*x^(2n)/faq(2n,q) and faq(n,q) = Product_{k=1..n} (q^k-1)/(q-1) is the q-factorial of n.

			Nonzero coefficients in row n range from x^(n-1) to x^(2n(n-1)) for n>0.
Triangle begins:
  1;
  1;
  0,1,2,1,1;
  0,0,1,3,5,8,10,10,9,7,5,2,1;
  0,0,0,1,4,10,21,36,55,78,101,122,138,145,143,134,117,95,72,50,32,18,9,3,1;
  0,0,0,0,1,5,16,41,87,164,283,452,679,967,1311,1700,2118,2540,2937,3282,3546,3706,3751,3676,3487,3202,2842,2436,2014,1602,1223,894,622,409,253,145,76,35,14,4,1;
  ...
Explicit expansion of g.f.:
1/cos_q(x,q) = 1 + x^2/faq(2,q) + x^4*(q + 2*q^2 + q^3 + q^4)/faq(4,q) +
x^6*(q^2 + 3*q^3 + 5*q^4 + 8*q^5 + 10*q^6 + 10*q^7 + 9*q^8 + 7*q^9 + 5*q^10 + 2*q^11 + q^12)/faq(6,q) +
x^8*(q^3 + 4*q^4 + 10*q^5 + 21*q^6 + 36*q^7 + 55*q^8 + 78*q^9 + 101*q^10 + 122*q^11 + 138*q^12 + 145*q^13 + 143*q^14 + 134*q^15 + 117*q^16 + 95*q^17 + 72*q^18 + 50*q^19 + 32*q^20 + 18*q^21 + 9*q^22 + 3*q^23 + q^24)/faq(8,q) +...

Paul D. Hanna, Table of n, a(n) for n = 0..2255, as a flattened triangle of rows 0..15
M. M. Graev, Einstein equations for invariant metrics on flag spaces and their Newton polytopes, Transactions of the Moscow Mathematical Society, 2014, pp. 13-68. Original publication: Trudy Moskovskogo Matematicheskogo Obshchestva, tom 75 (2014), vypusk 1.
Eric Weisstein, q-Cosine Function from MathWorld.
Eric Weisstein, q-Factorial from MathWorld.

Cf. A000364 (row sums=Euler numbers); A152801, A152802, A152803, A152804.

Cf. A152290, A152550.

{T(n,k)=polcoeff(polcoeff(1/sum(m=0,n,(-1)^m*x^(2*m)/prod(j=1,2*m,(q^j-1)/(q-1))+x*O(x^(2*n+1))),2*n,x)*prod(j=1,2*n,(q^j-1)/(q-1)),k,q)}
for(n=0,8,for(k=0,2*n*(n-1),print1(T(n,k),", "));print(""))

G.f.: 1/cos_q(x,q) = Sum_{n>=0} Sum_{k=0..2n(n-1)} T(n,k)*q^k*x^(2n)/faq(2n,q).
G.f.: 1/cos(x) = Sum_{n>=1} Sum_{k=0..2n(n-1)} T(n,k)*x^(2n)/(2n)!.
Sum_{k=0..2n(n-1)} T(n,k) = A000364(n).
Sum_{k=0..2n(n-1)} T(n,k)*(-1)^k = 1 for n>=0.
Sum_{k=0..2n(n-1)} T(n,k)*I^k = (-1)^[n/2] for n>=0 where I^2=-1.
Sum_{k=0..2n(n-1)} T(n,k)*exp(2*Pi*I*k/n) = 1 for n>0.

A152551 a(n) = (2*n+1)^floor((n-1)/2).

Original entry on oeis.org

1, 1, 1, 7, 9, 121, 169, 3375, 4913, 130321, 194481, 6436343, 9765625, 387420489, 594823321, 27512614111, 42618442977, 2251875390625, 3512479453921, 208728361158759, 327381934393961, 21611482313284249, 34050628916015625

Offset: 0

Paul D. Hanna, Dec 07 2008

nonn

Compare to row sums of triangle A152550: (2n+1)^(n-1).

Triangle A152550 lists coefficients in a q-analog of the function [LambertW(-2x)/(-2x)]^(1/2).

G. C. Greubel, Table of n, a(n) for n = 0..642

Cf. A152550, A152552 (q=2), A152553 (q=3).

[(2*n+1)^Floor((n-1)/2): n in [0..50]]; // G. C. Greubel, Nov 16 2017

Table[(2n+1)^Floor[(n-1)/2],{n,0,30}] (* Harvey P. Dale, Nov 21 2011 *)

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

a(n) = Sum_{k=0..n(n-1)/2} A152550(n,k)*(-1)^k.

A152552 Coefficients in a q-analog of the function [LambertW(-2x)/(-2x)]^(1/2) at q=2.

Original entry on oeis.org

1, 1, 7, 148, 7611, 872341, 213651052, 109327540680, 115381584785027, 249159124679346991, 1095244903267253760231, 9765839519517673327876328, 176188639876138769279299798900, 6419535615261099235478072782943388

Offset: 0

Paul D. Hanna, Dec 07 2008

nonn

			G.f.: A(x) = 1 + x + 7/3*x^2 + 148/21*x^3 + 7611/315*x^4 + 872341/9765*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. A152550, A152551 (q=-1), A152553 (q=3); A005329.

{a(n,q=2)=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(LW2_q+x*O(x^n),n,x)*prod(i=1,n,(q^i-1)/(q-1))}

G.f. satisfies: A(x) = e_q( x*A(x)^2, 2) and A( x/e_q(x,2)^2 ) = e_q(x,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 )]^(1/2)
a(n) = Sum_{k=0..n(n-1)/2} A152550(n,k)*2^k.

A152553 Coefficients in a q-analog of the function [LambertW(-2x)/(-2x)]^(1/2) at q=3.

Original entry on oeis.org

1, 1, 9, 339, 44521, 19059921, 25799597265, 108657870607875, 1410396873934264497, 56078100848527445045121, 6801233273726638573734096441, 2508450630100541880792088526933139

Offset: 0

Paul D. Hanna, Dec 07 2008

nonn

			G.f.: A(x) = 1 + x + 9/4*x^2 + 339/52*x^3 + 44521/2080*x^4 + 19059921/251680*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. A152550, A152551 (q=-1), A152552 (q=2); A015001.

{a(n,q=3)=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(LW2_q+x*O(x^n),n,x)*prod(i=1,n,(q^i-1)/(q-1))}

G.f. satisfies: A(x) = e_q( x*A(x)^2, 3) and A( x/e_q(x,3)^2 ) = e_q(x,3) 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 )]^(1/2)
a(n) = Sum_{k=0..n(n-1)/2} A152550(n,k)*3^k.

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A152554 Largest term in row n of triangle A152550.

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A152555 Coefficients in a q-analog of the function LambertW(-2*x)/(-2*x), as a triangle read by rows.

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A152800 Irregular triangle read by rows: the q-analog of the Euler numbers; expansion of the arithmetic inverse of the q-cosine of x.

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A152551 a(n) = (2*n+1)^floor((n-1)/2).

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A152552 Coefficients in a q-analog of the function [LambertW(-2x)/(-2x)]^(1/2) at q=2.

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A152553 Coefficients in a q-analog of the function [LambertW(-2x)/(-2x)]^(1/2) at q=3.

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A152555 Coefficients in a q-analog of the function LambertW(-2x)/(-2x), as a triangle read by rows.