This site is supported by donations to The OEIS Foundation.

 Hints (Greetings from The On-Line Encyclopedia of Integer Sequences!)
 A152800 The q-analog of the Euler numbers; expansion of the arithmetic inverse of the q-cosine of x. 5
 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 (list; graph; refs; listen; history; text; internal format)
 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 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 Non-zero 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)} CROSSREFS Cf. A000364 (row sums=Euler numbers); A152801, A152802, A152803, A152804. Cf. A152290, A152550. Sequence in context: A078807 A208249 A029422 * A223730 A117452 A029412 Adjacent sequences:  A152797 A152798 A152799 * A152801 A152802 A152803 KEYWORD nonn,tabf AUTHOR Paul D. Hanna, Dec 26 2008 STATUS approved

Lookup | Welcome | Wiki | Register | Music | Plot 2 | Demos | Index | Browse | More | WebCam
Contribute new seq. or comment | Format | Transforms | Puzzles | Hot | Classics
Recent Additions | More pages | Superseeker | Maintained by The OEIS Foundation Inc.

Content is available under The OEIS End-User License Agreement .