A152800 - OEIS

%I #21 Feb 16 2025 08:33:09

%S 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,

%T 101,122,138,145,143,134,117,95,72,50,32,18,9,3,1,0,0,0,0,1,5,16,41,

%U 87,164,283,452,679,967,1311,1700,2118,2540,2937,3282,3546,3706,3751,3676,3487

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

%C 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.

%H Paul D. Hanna, <a href="/A152800/b152800.txt">Table of n, a(n) for n = 0..2255, as a flattened triangle of rows 0..15</a>

%H M. M. Graev, <a href="http://dx.doi.org/10.1090/S0077-1554-2014-00235-1">Einstein equations for invariant metrics on flag spaces and their Newton polytopes</a>, Transactions of the Moscow Mathematical Society, 2014, pp. 13-68. Original publication: Trudy Moskovskogo Matematicheskogo Obshchestva, tom 75 (2014), vypusk 1.

%H Eric Weisstein, <a href="https://mathworld.wolfram.com/q-Cosine.html">q-Cosine Function</a> from MathWorld.

%H Eric Weisstein, <a href="https://mathworld.wolfram.com/q-Factorial.html">q-Factorial</a> from MathWorld.

%F 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).

%F G.f.: 1/cos(x) = Sum_{n>=1} Sum_{k=0..2n(n-1)} T(n,k)*x^(2n)/(2n)!.

%F Sum_{k=0..2n(n-1)} T(n,k) = A000364(n).

%F Sum_{k=0..2n(n-1)} T(n,k)*(-1)^k = 1 for n>=0.

%F Sum_{k=0..2n(n-1)} T(n,k)*I^k = (-1)^[n/2] for n>=0 where I^2=-1.

%F Sum_{k=0..2n(n-1)} T(n,k)*exp(2*Pi*I*k/n) = 1 for n>0.

%e Nonzero coefficients in row n range from x^(n-1) to x^(2n(n-1)) for n>0.

%e Triangle begins:

%e   1;

%e   1;

%e   0,1,2,1,1;

%e   0,0,1,3,5,8,10,10,9,7,5,2,1;

%e   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;

%e   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;

%e   ...

%e Explicit expansion of g.f.:

%e 1/cos_q(x,q) = 1 + x^2/faq(2,q) + x^4*(q + 2*q^2 + q^3 + q^4)/faq(4,q) +

%e 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) +

%e 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) +...

%o (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)}

%o for(n=0,8,for(k=0,2*n*(n-1),print1(T(n,k),", "));print(""))

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

%Y Cf. A152290, A152550.

%K nonn,tabf,changed

%O 0,5

%A _Paul D. Hanna_, Dec 26 2008