cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-5 of 5 results.

A152802 a(n) = A152800(n+2,2n+1) for n>=0.

Original entry on oeis.org

1, 3, 10, 41, 172, 749, 3332, 15041, 68640, 315840, 1462798, 6810588, 31846811, 149459541, 703592472, 3321019270, 15711717162, 74482623635, 353723268817, 1682536854931, 8014676326925, 38226681972410, 182538225520073
Offset: 0

Views

Author

Paul D. Hanna, Dec 26 2008

Keywords

Comments

Triangle A152800 gives a q-analog of the Euler numbers.

Links

Crossrefs

Cf. A152800; A152801.

Programs

  • PARI
    {a(n)=polcoeff(polcoeff(1/sum(m=0,n+2,(-1)^m*x^(2*m)/prod(j=1,2*m,(q^j-1)/(q-1))+x*O(x^(2*n+4))),2*n+4,x)*prod(j=1,2*n+4,(q^j-1)/(q-1)),2*n+1,q)}

A152801 a(n) = A152800(n+1,2n) for n>=0.

Original entry on oeis.org

1, 2, 5, 21, 87, 378, 1682, 7596, 34688, 159724, 740243, 3448579, 16134813, 75760294, 356811308, 1684882778, 7974184903, 37815280813, 179644013528, 854749753320, 4072634928710, 19429522879905, 92799859650401
Offset: 0

Views

Author

Paul D. Hanna, Dec 26 2008

Keywords

Comments

Triangle A152800 gives a q-analog of the Euler numbers.

Crossrefs

Cf. A152800; A152802.

Programs

  • PARI
    {a(n)=polcoeff(polcoeff(1/sum(m=0,n+1,(-1)^m*x^(2*m)/prod(j=1,2*m,(q^j-1)/(q-1))+x*O(x^(2*n+2))),2*n+2,x)*prod(j=1,2*n+2,(q^j-1)/(q-1)),2*n,q)}

A152803 G.f.: limit of the ratio of the g.f.s of adjacent rows in triangle A152800.

Original entry on oeis.org

1, 1, 2, 2, 2, 3, 4, 3, 3, 6, 6, 2, 5, 13, 5, -8, 17, 38, -28, -46, 119, 88, -290, -42, 763, -208, -1738, 1363, 3727, -5084, -6907, 16043, 9959, -45166, -4025, 116799, -44571, -277503, 244975, 597094, -912129, -1108463, 2880463, 1505334, -8180133, -30884
Offset: 1

Views

Author

Paul D. Hanna, Dec 26 2008

Keywords

Examples

			G.f.: q + q^2 + 2*q^3 + 2*q^4 + 2*q^5 + 3*q^6 + 4*q^7 + 3*q^8 + 3*q^9 + 6*q^10 +...

Crossrefs

Cf. A152800.

A152804 Largest term in rows of triangle A152800.

Original entry on oeis.org

1, 1, 2, 10, 145, 3751, 151646, 8867442, 705137822, 73336545762, 9646131627159, 1567174898585612, 307919420040219223, 72014984300625678228, 19759119083460510552459, 6288107535202467022813164
Offset: 0

Views

Author

Paul D. Hanna, Dec 26 2008

Keywords

Comments

Compare to the row sums of A152800, which is the Euler numbers (A000364).

Crossrefs

Cf. A152800.

A152805 Expansion of 2/(1 + e_q(2x,q)) where e_q(2x,q) is the q-exponential of 2x, as a triangle of coefficients read by rows.

Original entry on oeis.org

1, -1, -1, 1, -1, 2, 2, -1, -1, 3, 3, 0, -3, -3, 1, -1, 4, 3, -3, -4, -14, -4, -3, 3, 4, -1, -1, 5, 2, -9, -11, -19, -16, -11, 11, 16, 19, 11, 9, -2, -5, 1, -1, 6, 0, -17, -18, -25, 1, -7, 41, 73, 83, 83, 73, 41, -7, 1, -25, -18, -17, 0, 6, -1, -1, 7, -3, -26, -20, -17, 38, 67
Offset: 0

Views

Author

Paul D. Hanna, Dec 28 2008

Keywords

Comments

May be considered as coefficients of a q-analog of the tangent numbers (A000182).
The q-exponential of x is e_q(x,q) = Sum_{n>=0} x^n/faq(n,q) where faq(n,q) = Product_{k=1..n} (q^k-1)/(q-1) is the q-factorial of n.

Examples

			Row n lists coefficients of powers of q ranging from q^0 to q^(n(n-1)/2).
Triangle begins:
1;
-1;
-1,1;
-1,2,2,-1;
-1,3,3,0,-3,-3,1;
-1,4,3,-3,-4,-14,-4,-3,3,4,-1;
-1,5,2,-9,-11,-19,-16,-11,11,16,19,11,9,-2,-5,1;
-1,6,0,-17,-18,-25,1,-7,41,73,83,83,73,41,-7,1,-25,-18,-17,0,6,-1;
-1,7,-3,-26,-20,-17,38,67,115,184,223,217,198,84,0,-84,-198,-217,-223,-184,-115,-67,-38,17,20,26,3,-7,1;
-1,8,-7,-35,-13,12,110,161,258,271,261,219,33,-257,-638,-876,-1269,-1423,-1564,-1423,-1269,-876,-638,-257,33,219,261,271,258,161,110,12,-13,-35,-7,8,-1;
...
EXPLICIT EXPANSION OF G.F.:
1 - x + x^2*(-1 + q)/faq(2,q) + x^3*(-1 + 2*q + 2*q^2 - q^3)/faq(3,q) +
x^4*(-1 + 3*q + 3*q^2 - 3*q^4 - 3*q^5 + q^6)/faq(4,q) +
x^5*(-1 + 4*q + 3*q^2 - 3*q^3 - 4*q^4 - 14*q^5 - 4*q^6 - 3*q^7 + 3*q^8 + 4*q^9 - q^10)/faq(5,q) +...

Crossrefs

Cf. A000182 (row sums=signed tangent numbers); A152806 (unsigned row sums); A152807; A152800.

Programs

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

Formula

Sum_{k=0..n(n-1)/2} T(n,k) * exp(2*Pi*I*k/n) = -2^(n-1) for n>0.
Showing 1-5 of 5 results.

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

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