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

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

0,5

Comments

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.

Links

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.

Formula

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.

Example

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;
0,0,0,0,0,1,6,23,69,172,378,754,1390,2404,3938,6153,9223,13323,18609,25203,33174,42514,53130,64834,77336,90255,103136,115470,126726,136390,143998,149170,151646,151299,148146,142351,134207,124115,112555,100050,87126,74281,61955,50504,40192,31187,23556,17286,12297,8456,5601,3558,2155,1235,664,330,149,59,20,5,1;
...
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) +...

Prog

(PARI) {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(""))

Crossrefs

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

Cf. A152290, A152550.

Sequence in context: A329985 A029422 A351356 * A223730 A353129 A117452

Adjacent sequences: A152797 A152798 A152799 * A152801 A152802 A152803

Keyword

nonn,tabf

Author

Paul D. Hanna, Dec 26 2008

Status

approved