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 1-3 of 3 results.

A322188 G.f.: exp( Sum_{n>=1} A322187(n)*x^n/n ), where A322187(n) is the coefficient of x^n*y^n/n in log( Product_{n>=1} 1/(1 - x^(2*n-1) - y^(2*n-1)) ).

Original entry on oeis.org

1, 1, 2, 6, 15, 45, 140, 448, 1483, 5027, 17311, 60469, 213678, 762284, 2741864, 9932346, 36202666, 132677658, 488605698, 1807176452, 6710206574, 25003642942, 93468147306, 350425771854, 1317330452697, 4964398631867, 18751217069083, 70975750129731, 269180061675328, 1022750160098864, 3892577330120307, 14838784128136803, 56651259287153670, 216586672901518164, 829142137823283601, 3178107527615273349
Offset: 0

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Author

Paul D. Hanna, Dec 07 2018

Keywords

Examples

			G.f.: A(x) = 1 + x + 2*x^2 + 6*x^3 + 15*x^4 + 45*x^5 + 140*x^6 + 448*x^7 + 1483*x^8 + 5027*x^9 + 17311*x^10 + 60469*x^11 + 213678*x^12 + ...
such that
log(A(x)) = x + 3*x^2/2 + 13*x^3/3 + 35*x^4/4 + 131*x^5/5 + 471*x^6/6 + 1723*x^7/7 + 6435*x^8/8 + 24349*x^9/9 + 92393*x^10/10 + 352727*x^11/11 + 1352183*x^12/12 + ... + A322187(n)*x^n/n + ...
RELATED SERIES.
A(x)^2 = 1 + 2*x + 5*x^2 + 16*x^3 + 46*x^4 + 144*x^5 + 466*x^6 + 1536*x^7 + 5187*x^8 + 17842*x^9 + 62209*x^10 + 219504*x^11 + 782272*x^12 + ...

Links

Crossrefs

Cf. A322187, A322198, A322204.

Programs

  • PARI
    N=35;
{L = sum(n=1, N+1, -log(1 - x^(2*n-1) - y^(2*n-1) +x*O(x^N) +y*O(y^N)) ); }
{A322187(n) = polcoeff( n*polcoeff( L, n, x), n, y)}
{a(n) = polcoeff( exp( sum(m=1, n, A322187(m)*x^m/m ) +x*O(x^n) ), n) }
for(n=0, N, print1( a(n), ", ") )

Formula

a(n) ~ c * 4^n / n^(3/2), where c = 0.57389010009720382786456367148681469430628117317... - Vaclav Kotesovec, Jun 18 2019

A322198 a(n) is the coefficient of x^n*y^n in Product_{n>=1} 1/(1 - x^(2*n-1) - y^(2*n-1)).

Original entry on oeis.org

1, 2, 6, 24, 84, 312, 1174, 4420, 16772, 64014, 245212, 942668, 3634914, 14051530, 54440336, 211331906, 821779372, 3200447054, 12481364146, 48736064248, 190513382908, 745492958862, 2919891150694, 11446207136530, 44905452622268, 176300343498632, 692629144937724, 2722834581642342, 10710164125130394, 42151077430686344, 165975440541202824, 653864689092828458
Offset: 0

Views

Author

Paul D. Hanna, Dec 07 2018

Keywords

Examples

			G.f.: A(x) = 1 + 2*x + 6*x^2 + 24*x^3 + 84*x^4 + 312*x^5 + 1174*x^6 + 4420*x^7 + 16772*x^8 + 64014*x^9 + 245212*x^10 + 942668*x^11 + 3634914*x^12 + ...
RELATED SERIES.
The product P(x,y) = Product_{n>=1} 1/(1 - x^(2*n-1) - y^(2*n-1)) = 1/( (1 - x - y) * (1 - x^3 - y^3) * (1 - x^5 - y^5) * (1 - x^7 - y^7) * (1 - x^9 - y^9) * ...)
may be expressed as the series expansion
P(x,y) = 1 + (1*x + 1*y) + (1*x^2 + 2*x*y + 1*y^2) + (2*x^3 + 3*x^2*y + 3*x*y^2 + 2*y^3) + (2*x^4 + 5*x^3*y + 6*x^2*y^2 + 5*x*y^3 + 2*y^4) + (3*x^5 + 7*x^4*y + 11*x^3*y^2 + 11*x^2*y^3 + 7*x*y^4 + 3*y^5) + (4*x^6 + 10*x^5*y + 18*x^4*y^2 + 24*x^3*y^3 + 18*x^2*y^4 + 10*x*y^5 + 4*y^6) + (5*x^7 + 14*x^6*y + 28*x^5*y^2 + 42*x^4*y^3 + 42*x^3*y^4 + 28*x^2*y^5 + 14*x*y^6 + 5*y^7) + (6*x^8 + 19*x^7*y + 42*x^6*y^2 + 71*x^5*y^3 + 84*x^4*y^4 + 71*x^3*y^5 + 42*x^2*y^6 + 19*x*y^7 + 6*y^8) + ...
in which this sequence equals the coefficients of x^n*y^n for n >= 0.

Links

Crossrefs

Cf. A322187, A322211.

Programs

  • PARI
    N=35;
{P = 1/prod(n=1, N+1, (1 - x^(2*n-1) - y^(2*n-1) +x^2*O(x^N) +y^2*O(y^N)) ); }
{a(n) = polcoeff( polcoeff( P, n, x), n, y)}
for(n=0, N, print1( a(n), ", ") )

Formula

a(n) ~ c * 4^n / sqrt(n), where c = 1/(sqrt(Pi) * QPochhammer(1/4)) = 0.819402796697705077405540985476846791094716961849197... - Vaclav Kotesovec, Jun 18 2019, updated Mar 17 2024

A322185 a(n) = sigma(2*n) * binomial(2*n,n)/2, for n >= 1.

Original entry on oeis.org

3, 21, 120, 525, 2268, 12936, 41184, 199485, 948090, 3879876, 12697776, 81124680, 218412600, 1123264800, 5584230720, 18934032285, 63007367940, 412918656150, 1060357914000, 6203093796900, 25836377973120, 88372156476240, 296403506193600, 1999351428352200, 5878093199355468, 24300008114457096, 116816365538886720, 458921436045626400, 1353026992479346800
Offset: 1

Views

Author

Paul D. Hanna, Dec 07 2018

Keywords

Comments

Related logarithmic series:
(1) log( Product_{n>=1} (1 - x^(2*n))/(1 - x^n)^3 ) = Sum_{n>=1} sigma(2*n) * x^n/n (see formula of Joerg Arndt in A182818).
(2) log( C(x) ) = Sum_{n>=1} binomial(2*n,n)/2 * x^n/n, where C(x) = 1 + x*C(x)^2 is the Catalan function (A000108).

Examples

			L.g.f: L(x) = 3*x + 21*x^2/2 + 120*x^3/3 + 525*x^4/4 + 2268*x^5/5 + 12936*x^6/6 + 41184*x^7/7 + 199485*x^8/8 + 948090*x^9/9 + 3879876*x^10/10 + 12697776*x^11/11 + ... + sigma(2*n) * binomial(2*n,n)/2 * x^n/n + ...
RELATED SERIES.
exp(L(x)) = 1 + 3*x + 15*x^2 + 76*x^3 + 357*x^4 + 1662*x^5 + 8203*x^6 + 36609*x^7 + 169800*x^8 + 788024*x^9 + 3586350*x^10 + 15948147*x^11 + ... + A322186(n)*x^n + ...
The table of coefficients of x^n*y^k/(n+k) in
log( Product_{n>=1} 1/(1 - (x + y)^n) ) = (1*x + 1*y)/1 + (3*x^2 + 6*x*y + 3*y^2)/2 + (4*x^3 + 12*x^2*y + 12*x*y^2 + 4*y^3)/3 + (7*x^4 + 28*x^3*y + 42*x^2*y^2 + 28*x*y^3 + 7*y^4)/4 + (6*x^5 + 30*x^4*y + 60*x^3*y^2 + 60*x^2*y^3 + 30*x*y^4 + 6*y^5)/5 + (12*x^6 + 72*x^5*y + 180*x^4*y^2 + 240*x^3*y^3 + 180*x^2*y^4 + 72*x*y^5 + 12*y^6)/6 + (8*x^7 + 56*x^6*y + 168*x^5*y^2 + 280*x^4*y^3 + 280*x^3*y^4 + 168*x^2*y^5 + 56*x*y^6 + 8*y^7)/7 + (15*x^8 + 120*x^7*y + 420*x^6*y^2 + 840*x^5*y^3 + 1050*x^4*y^4 + 840*x^3*y^5 + 420*x^2*y^6 + 120*x*y^7 + 15*y^8)/8 + ...
begins
n=0: [0, 1, 3, 4, 7, 6, 12, 8, 15, 13, 18, ..., sigma(k), ...];
n=1: [1, 6, 12, 28, 30, 72, 56, 120, 117, 180, ...];
n=2: [3, 12, 42, 60, 180, 168, 420, 468, 810, 660, ...];
n=3: [4, 28, 60, 240, 280, 840, 1092, 2160, 1980, 6160, ...];
n=4: [7, 30, 180, 280, 1050, 1638, 3780, 3960, 13860, 10010, ...];
n=5: [6, 72, 168, 840, 1638, 4536, 5544, 22176, 18018, 48048, ...];
n=6: [12, 56, 420, 1092, 3780, 5544, 25872, 24024, 72072, 120120, ...];
n=7: [8, 120, 468, 2160, 3960, 22176, 24024, 82368, 154440, 354640, ...];
n=8: [15, 117, 810, 1980, 13860, 18018, 72072, 154440, 398970, 437580, ...];
n=9: [13, 180, 660, 6160, 10010, 48048, 120120, 354640, 437580, 1896180, ...];
n=10: [18, 132, 1848, 4004, 24024, 72072, 248248, 350064, 1706562, 1847560, ...]; ...
in which the diagonal of coefficients of x^n*y^n/(2*n) equals
[0, 6, 42, 240, 1050, 4536, 25872, 82368, 398970, 1896180, ..., 2*a(n), ...],
which is twice this sequence.

Links

Crossrefs

Cf. A322186, A322203, A322187.

Programs

  • PARI
    {a(n) = sigma(2*n) * binomial(2*n,n)/2}
for(n=1, 30, print1( a(n), ", ") )
  • PARI
    /* [x^n*y^n/n] log( Product_{n>=1} 1/(1 - (x + y)^n) ) */
N=30
{L = sum(n=1, 2*N+1, -log(1 - (x + y)^n +x*O(x^(2*N)) +y*O(y^(2*N))) ); }
{a(n) = polcoeff( n*polcoeff( L, n, x), n, y)}
for(n=1, N, print1( a(n), ", ") )

Formula

a(n) is the coefficient of x^n*y^n/n in log( Product_{n>=1} 1/(1 - (x + y)^n) ), for n >= 1.
Showing 1-3 of 3 results.

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