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 1-4 of 4 results.

A161801 G.f. is Q_0(q) where Q_0(q^4) is a series quadrisection of the g.f. of A161800.

Original entry on oeis.org

1, -6, -8, 112, -86, -752, 1360, 1216, -5384, 10762, -8176, -59888, 130160, 47696, -306336, 485952, -632982, -1582304, 4638088, 343120, -6514672, 8034464, -16636656, -20670528, 82724176, 17877578, -114481936, 52539968, -178638656
Offset: 0

Views

Author

Paul D. Hanna, Jul 19 2009

Keywords

Comments

The g.f. of A161800 has two nonzero series quadrisections; the other is given by A161802.

Examples

			G.f.: Q_0(q) = 1 - 6*q - 8*q^2 + 112*q^3 - 86*q^4 - 752*q^5 + 1360*q^6 +...

Crossrefs

Cf. A161800, A161802.

Programs

  • PARI
    {a(n)=local(L=sum(m=1, 4*n,2*2^valuation(m,2)*sumdiv(m, d, -(-1)^d*d)*x^m/m)+x*O(x^(4*n))); polcoeff(exp(L), 4*n)}

A161802 G.f. is Q_1(q) where q*Q_1(q^4) is a series quadrisection of the g.f. of A161800.

Original entry on oeis.org

2, -16, 18, 176, -544, -160, 2834, -5104, 3232, 18032, -68992, 48400, 143074, -343088, 461344, 63888, -2298880, 2963520, 1387424, -5145536, 10416514, -9297312, -31084704, 42991712, 34760672, -51170800, 81567168, -94111088
Offset: 0

Views

Author

Paul D. Hanna, Jul 19 2009

Keywords

Comments

The g.f. of A161800 has two nonzero series quadrisections; the other is given by A161801.

Examples

			G.f.: Q_1(q) = 2 - 16*q + 18*q^2 + 176*q^3 - 544*q^4 - 160*q^5 + 2834*q^6 +...

Crossrefs

Cf. A161800, A161801.

Programs

  • PARI
    {a(n)=local(L=sum(m=1, 4*n+1,2*2^valuation(m,2)*sumdiv(m, d, -(-1)^d*d)*x^m/m)+x*O(x^(4*n+1))); polcoeff(exp(L), 4*n+1)}

A161804 G.f.: A(q) = exp( Sum_{n>=1} A002129(n) * 3*A038500(n) * q^n/n ).

Original entry on oeis.org

1, 3, 3, 12, 30, 27, 66, 141, 111, 255, 513, 378, 903, 1815, 1356, 2970, 5727, 4131, 8571, 15882, 10881, 23001, 42417, 29106, 59763, 108165, 73500, 145164, 255831, 167643, 333693, 585258, 382053, 751059, 1302966, 849339, 1623009, 2762349
Offset: 0

Views

Author

Paul D. Hanna, Jul 20 2009

Keywords

Comments

A002129 forms the l.g.f. of log[ Sum_{n>=0} x^(n(n+1)/2) ], while 3*A038500 forms the l.g.f. of the log of the g.f. of A161809 and A038500(n) is the highest power of 3 dividing n.

Examples

			G.f.: A(q) = 1 + 3*q + 3*q^2 + 12*q^3 + 30*q^4 + 27*q^5 + 66*q^6 + ...
log(A(q)) = 3*q - 3*q^2 + 36*q^3 - 15*q^4 + 18*q^5 - 36*q^6 + 24*q^7 + ...
Sum_{n>=1} A002129(n)*q^n/n = log(1 + q + q^3 + q^6 + q^10 + q^15 + ...),
Sum_{n>=1} 3*A038500(n)*x^n/n = log of the g.f. of A161809.
TRISECTIONS:
T_0(q) = 1 + 12*q + 66*q^2 + 255*q^3 + 903*q^4 + 2970*q^5 + ... (A161805)
T_1(q) = 3 + 30*q + 141*q^2 + 513*q^3 + 1815*q^4 + 5727*q^5 + ... (A161806)
T_2(q) = 3 + 27*q + 111*q^2 + 378*q^3 + 1356*q^4 + 4131*q^5 + ... (A161807)
where T_1(-q)/T_0(-q)/3 equals (cf. A132977):
1 + 2*q + 5*q^2 + 12*q^3 + 26*q^4 + 50*q^5 + 92*q^6 + 168*q^7 + ...
and T_2(-q)/T_0(-q)/3 equals (cf. A132978):
1 + 3*q + 7*q^2 + 15*q^3 + 32*q^4 + 63*q^5 + 114*q^6 + 201*q^7 + ...
also, T_2(q)/T_1(q) equals (cf. A092848):
1 - q + 2*q^3 - 2*q^4 - q^5 + 4*q^6 - 4*q^7 - q^8 + 8*q^9 - 8*q^10 + ...

Crossrefs

Cf. trisections: A161805 (T_0), A161806 (T_1), A161807 (T_2).
Cf. A132977 (T_1/T_0), A132978 (T_2/T_0), A092848 (T_2/T_1).
Cf. A002129, A038500, A161809, A161800 (variant).

Programs

  • PARI
    {a(n)=local(L=sum(m=1, n,3*3^valuation(m,3)*sumdiv(m, d, -(-1)^d*d)*x^m/m)+x*O(x^n)); polcoeff(exp(L), n)}

Formula

Given trisections where A(q) = T_0(q^3) + q*T_1(q^3) + q^2*T_2(q^3):
T_0(q) = Sum_{n>=0} a(3n)*q^n,
T_1(q) = Sum_{n>=0} a(3n+1)*q^n,
T_2(q) = Sum_{n>=0} a(3n+2)*q^n,
then it appears that:
T_1(-q)/T_0(-q) = 3*q^(-1/3)*(eta(q^6)^4/(eta(q)*eta(q^3)*eta(q^4)*eta(q^12)))^2 (Cf. A132977);
T_2(-q)/T_0(-q) = 3*q^(-2/3)*(eta(q^2)*eta(q^6))^2*eta(q^3)*eta(q^12)/(eta(q)*eta(q^4))^3 (cf. A132978);
T_2(q)/T_1(q) = g.f. of A092848, the reciprocal of Hauptmodul for Gamma_0(18).

A161803 G.f.: A(x) = exp( Sum_{n>=1} A162552(n) * 2*A006519(n) * x^n/n ).

Original entry on oeis.org

1, 2, 0, -2, 6, 12, 0, -8, 24, 44, 0, -30, 54, 104, 0, -60, 238, 466, 0, -402, 924, 1892, 0, -1228, 3264, 6006, 0, -4052, 6688, 13052, 0, -7452, 16536, 32140, 0, -24828, 39660, 85744, 0, -53592, 114336, 212406, 0, -141090, 190754, 386956, 0, -216572, 136078
Offset: 0

Views

Author

Paul D. Hanna, Jul 19 2009

Keywords

Comments

A162552 forms the l.g.f. of log[ Sum_{n>=0} x^(n^2) ], while
2*A006519 forms the l.g.f. of binary partitions (A000123) and
A006519(n) is the highest power of 2 dividing n.

Examples

			G.f.: 1 + 2*x - 2*x^3 + 6*x^4 + 12*x^5 - 8*x^7 + 24*x^8 + 44*x^9 +...

Crossrefs

Cf. A161800, A162552.

Programs

  • PARI
    {a(n)=local(SQ=sum(m=0, sqrtint(n+1), x^(m^2))+x*O(x^n), L=sum(m=1,n,2*2^valuation(m,2)*polcoeff(log(SQ),m)*x^m)+x*O(x^n)); polcoeff(exp(L),n)}
Showing 1-4 of 4 results.

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