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.

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A322204 G.f.: exp( Sum_{n>=1} A322203(n)*x^n/n ), where A322203(n) is the coefficient of x^n*y^n/n in Sum_{n>=1} -log(1 - (x^n + y^n)).

Original entry on oeis.org

1, 1, 3, 7, 20, 54, 168, 518, 1702, 5672, 19413, 67329, 236994, 842362, 3022320, 10924142, 39749219, 145457241, 534996370, 1976582432, 7332199623, 27298096431, 101968071485, 382033462335, 1435270419582, 5405847465772, 20408264704999, 77211968620103, 292706146651697, 1111698968597495, 4229571286335997, 16117966287887641, 61515492682026560, 235114188287816030, 899821838980825557, 3448133313264656915
Offset: 0

Views

Author

Paul D. Hanna, Nov 30 2018

Keywords

Comments

Conjecture: Euler transform of A003239. - Georg Fischer, Dec 10 2020

Examples

			G.f.: A(x) = 1 + x + 3*x^2 + 7*x^3 + 20*x^4 + 54*x^5 + 168*x^6 + 518*x^7 + 1702*x^8 + 5672*x^9 + 19413*x^10 + 67329*x^11 + 236994*x^12 + ...
such that
log( A(x) ) = x + 5*x^2/2 + 13*x^3/3 + 45*x^4/4 + 131*x^5/5 + 497*x^6/6 + 1723*x^7/7 + 6525*x^8/8 + 24349*x^9/9 + 92655*x^10/10 + ... + A322203(n)*x^n/n + ...
Also,
A(x)^2  = 1 + 2*x + 7*x^2 + 20*x^3 + 63*x^4 + 190*x^5 + 613*x^6 + 1976*x^7 + 6604*x^8 + 22368*x^9 + 77270*x^10 + 270208*x^11 + 956780*x^12 + ... + A322202(n)*x^n + ...

Links

Crossrefs

Cf. A000108, A003239, A322200, A322203, A322202, A309682.

Programs

  • Maple
    C:= proc(n) option remember; binomial(2*n, n)/(n+1) end:
b:= proc(n, i) option remember; `if`(n=0 or i=1, C(n),
      add((t-> b(t, min(t, i-1)))(n-i*j)*C(j), j=0..n/i))
    end:
a:= n-> b(n$2):
seq(a(n), n=0..35);  # Alois P. Heinz, Aug 24 2019
  • Mathematica
    nmax = 25; CoefficientList[Series[Product[Sum[CatalanNumber[k]*x^(j*k), {k, 0, nmax/j}], {j, 1, nmax}], {x, 0, nmax}], x] (* Vaclav Kotesovec, Aug 12 2019 *)
nmax = 25; CoefficientList[Series[Product[(1 - Sqrt[1 - 4*x^k])/(2*x^k), {k, 1, nmax}], {x, 0, nmax}], x] (* Vaclav Kotesovec, Aug 12 2019 *)
  • PARI
    {L = sum(n=1,61, -log(1 - (x^n + y^n) +O(x^61) +O(y^61)) );}
{A322203(n) = polcoeff( n*polcoeff( L,n,x),n,y)}
{a(n) = polcoeff( exp( sum(m=1,n, A322203(m)*x^m/m ) +x*O(x^n) ),n) }
for(n=0,35, print1( a(n),", ") )

Formula

a(n) ~ c * 4^n / n^(3/2), where c = 1/sqrt(Pi) * Product_{j>=1} (2^(j+1) * (2^j - sqrt(4^j - 1))) = 0.6176761088360252844346512553859... - Vaclav Kotesovec, Jun 18 2019, updated Aug 12 2019
G.f.: Product_{j>=1} c(x^j), where c(x) = (1-sqrt(1-4*x))/(2*x) is the g.f. of A000108. - Alois P. Heinz, Aug 24 2019

A322200 L.g.f.: L(x,y) = log( Product_{n>=1} 1/(1 - (x^n + y^n)) ), where L(x,y) = Sum_{n>=0} Sum_{k>=0} T(n,k) * x^n*y^k / (n+k) such that L(0,0) = 0, as a symmetric square table of coefficients T(n,k) read by antidiagonals starting with T(0,0) = 0.

Original entry on oeis.org

0, 1, 1, 3, 2, 3, 4, 3, 3, 4, 7, 4, 10, 4, 7, 6, 5, 10, 10, 5, 6, 12, 6, 21, 26, 21, 6, 12, 8, 7, 21, 35, 35, 21, 7, 8, 15, 8, 36, 56, 90, 56, 36, 8, 15, 13, 9, 36, 93, 126, 126, 93, 36, 9, 13, 18, 10, 55, 120, 230, 262, 230, 120, 55, 10, 18, 12, 11, 55, 165, 330, 462, 462, 330, 165, 55, 11, 12, 28, 12, 78, 232, 537, 792, 994, 792, 537, 232, 78, 12, 28, 14, 13, 78, 286, 715, 1287, 1716, 1716, 1287, 715, 286, 78, 13, 14, 24, 14, 105, 364, 1043, 2002, 3073, 3446, 3073, 2002, 1043, 364, 105, 14, 24, 24, 15, 105, 470, 1365, 3018, 5035, 6435, 6435, 5035, 3018, 1365, 470, 105, 15, 24, 31, 16, 136, 560, 1892, 4368, 8120, 11440, 13050, 11440, 8120, 4368, 1892, 560, 136, 16, 31
Offset: 0

Views

Author

Paul D. Hanna, Nov 30 2018

Keywords

Examples

			L.g.f.: L(x,y) = (x + y)/1 + (3*x^2 + 2*x*y + 3*y^2)/2 + (4*x^3 + 3*x^2*y + 3*x*y^2 + 4*y^3)/3 + (7*x^4 + 4*x^3*y + 10*x^2*y^2 + 4*x*y^3 + 7*y^4)/4 + (6*x^5 + 5*x^4*y + 10*x^3*y^2 + 10*x^2*y^3 + 5*x*y^4 + 6*y^5)/5 + (12*x^6 + 6*x^5*y + 21*x^4*y^2 + 26*x^3*y^3 + 21*x^2*y^4 + 6*x*y^5 + 12*y^6)/6 + (8*x^7 + 7*x^6*y + 21*x^5*y^2 + 35*x^4*y^3 + 35*x^3*y^4 + 21*x^2*y^5 + 7*x*y^6 + 8*y^7)/7 + (15*x^8 + 8*x^7*y + 36*x^6*y^2 + 56*x^5*y^3 + 90*x^4*y^4 + 56*x^3*y^5 + 36*x^2*y^6 + 8*x*y^7 + 15*y^8)/8 + ...
such that
exp( L(x,y) ) = Product_{n>=1} 1/(1 - (x^n + y^n)), or
L(x,y) = Sum_{n>=1} -log(1 - (x^n + y^n)),
where
L(x,y) = Sum_{n>=0} Sum_{k>=0} T(n,k) * x^n*y^k/(n+k),
in which the constant term is taken to be zero: L(0,0) = 0.
SQUARE TABLE.
The square table of coefficients T(n,k) of x^n*y^k/(n+k) in L(x,y) begins
0, 1, 3, 4, 7, 6, 12, 8, 15, 13, 18, 12, 28, 14, ...;
1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, ...;
3, 3, 10, 10, 21, 21, 36, 36, 55, 55, 78, 78, 105, ...;
4, 4, 10, 26, 35, 56, 93, 120, 165, 232, 286, 364, ...;
7, 5, 21, 35, 90, 126, 230, 330, 537, 715, 1043, 1365, ...;
6, 6, 21, 56, 126, 262, 462, 792, 1287, 2002, 3018, ...;
12, 7, 36, 93, 230, 462, 994, 1716, 3073, 5035, 8120, ...;
8, 8, 36, 120, 330, 792, 1716, 3446, 6435, 11440, 19448, ...;
15, 9, 55, 165, 537, 1287, 3073, 6435, 13050, 24310, 44010, ...;
13, 10, 55, 232, 715, 2002, 5035, 11440, 24310, 48698, 92378, ...;
18, 11, 78, 286, 1043, 3018, 8120, 19448, 44010, 92378, 185310, ...;
12, 12, 78, 364, 1365, 4368, 12376, 31824, 75582, 167960, 352716, ...; ...
TRIANGLE.
Alternatively, this sequence may be written as a triangle, starting as
0;
1, 1;
3, 2, 3;
4, 3, 3, 4;
7, 4, 10, 4, 7;
6, 5, 10, 10, 5, 6;
12, 6, 21, 26, 21, 6, 12;
8, 7, 21, 35, 35, 21, 7, 8;
15, 8, 36, 56, 90, 56, 36, 8, 15;
13, 9, 36, 93, 126, 126, 93, 36, 9, 13;
18, 10, 55, 120, 230, 262, 230, 120, 55, 10, 18;
12, 11, 55, 165, 330, 462, 462, 330, 165, 55, 11, 12;
28, 12, 78, 232, 537, 792, 994, 792, 537, 232, 78, 12, 28;
14, 13, 78, 286, 715, 1287, 1716, 1716, 1287, 715, 286, 78, 13, 14;
24, 14, 105, 364, 1043, 2002, 3073, 3446, 3073, 2002, 1043, 364, 105, 14, 24;
24, 15, 105, 470, 1365, 3018, 5035, 6435, 6435, 5035, 3018, 1365, 470, 105, 15, 24;
31, 16, 136, 560, 1892, 4368, 8120, 11440, 13050, 11440, 8120, 4368, 1892, 560, 136, 16, 31; ...
where L(x,y) = Sum_{n>=0} Sum_{k=0..n} T(n-k,k)*x^(n-k)*y^k / n.

Links

Crossrefs

Cf. A322210 (exp), A322201 (main diagonal), A322203, A322205, A322207, A322209.
Cf. A054598 (antidiagonal sums), A054599.

Programs

  • PARI
    {L = sum(n=1,61, -log(1 - (x^n + y^n) +O(x^61) +O(y^61)) );}
{T(n,k) = polcoeff( (n+k)*polcoeff( L,n,x),k,y)}
for(n=0,16, for(k=0,16, print1( T(n,k),", ") );print(""))

Formula

Sum_{k=0..n} T(n-k,k) = A054598(n) = Sum_{d|n} d*2^(n/d).
Sum_{k=0..n} T(n-k,k) * k/n = A054599(n) = Sum_{d|n} d*2^(n/d - 1).
Sum_{k=0..n} T(n-k,k) * 2^k = A322209(n) = [x^n] log( Product_{k>=1} 1/(1 - (2^k+1)*x^k) ) for n >= 0.
FORMULAS FOR TERMS.
T(n,k) = T(k,n) for n >= 0, k >= 0.
T(0,0) = 0.
T(n,0) = sigma(n) for n > 0.
T(0,k) = sigma(k) for n > 0.
T(n,1) = n+1, for n >= 0.
T(1,k) = k+1, for k >= 0.
T(2*n,2) = T(2*n+1,2) = (n+1)*(2*n+3).
T(2,2*k) = T(2,2*k+1) = (k+1)*(2*k+3).
COLUMN GENERATING FUNCTIONS.
Row 0: log(P(x)), where P(x) = Product_{n>=1} 1/(1 - x^n).
Row 1: 1/(1-x)^2.
Row 2: (3 + x^2)/((1-x)*(1-x^2)^2).
Row 3: (4 - 4*x + 6*x^2 + 2*x^3 + x^4)/((1-x)^2*(1-x^3)^2).
Row 4: (7 - 9*x + 11*x^2 + 7*x^3 + 9*x^4 + x^5 + 5*x^6 + x^7)/((1-x)^2*(1-x^2)*(1-x^4)^2).
Row 5: (6 - 18*x + 33*x^2 - 16*x^3 + 10*x^4 + 4*x^5 + 3*x^6 + 2*x^7 + x^8)/((1-x)^3*(1-x^5)^2).
Row 6: (12 - 41*x + 56*x^2 + 13*x^3 - 49*x^4 - 20*x^5 + 105*x^6 - 126*x^7 + 85*x^8 - 62*x^9 + 24*x^10 - 28*x^11 + 39*x^12 - 25*x^13 + 15*x^14 + x^15 + x^16) / ((1-x)^4*(1-x^2)^2*(1-x^3)*(1-x^6)^2).

A322187 a(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)) ), for n >= 1.

Original entry on oeis.org

1, 3, 13, 35, 131, 471, 1723, 6435, 24349, 92393, 352727, 1352183, 5200313, 20058321, 77559203, 300540195, 1166803127, 4537569063, 17672631919, 68923264585, 269128942459, 1052049481893, 4116715363823, 16123801860855, 63205303219531, 247959266474091, 973469712897103, 3824345300380465, 15033633249770549, 59132290782710201, 232714176627630575, 916312070471295267, 3609714217009191161, 14226520737620288421
Offset: 1

Views

Author

Paul D. Hanna, Dec 07 2018

Keywords

Examples

			L.g.f.: L(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 + ...
RELATED SERIES.
exp(L(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 + ... + A322188(n)*x^n + ...
The table of coefficients of x^n*y^k/(n+k) in
log( Product_{n>=1} 1/(1 - x^(2*n-1) - y^(2*n-1)) ) = (1*x + 1*y) + (1*x^2 + 2*x*y + 1*y^2)/2 + (4*x^3 + 3*x^2*y + 3*x*y^2 + 4*y^3)/3 + (1*x^4 + 4*x^3*y + 6*x^2*y^2 + 4*x*y^3 + 1*y^4)/4 + (6*x^5 + 5*x^4*y + 10*x^3*y^2 + 10*x^2*y^3 + 5*x*y^4 + 6*y^5)/5 + (4*x^6 + 6*x^5*y + 15*x^4*y^2 + 26*x^3*y^3 + 15*x^2*y^4 + 6*x*y^5 + 4*y^6)/6 + (8*x^7 + 7*x^6*y + 21*x^5*y^2 + 35*x^4*y^3 + 35*x^3*y^4 + 21*x^2*y^5 + 7*x*y^6 + 8*y^7)/7 + (1*x^8 + 8*x^7*y + 28*x^6*y^2 + 56*x^5*y^3 + 70*x^4*y^4 + 56*x^3*y^5 + 28*x^2*y^6 + 8*x*y^7 + 1*y^8)/8 + ...
begins
n=0: [0, 1, 1, 4, 1, 6, 4, 8, 1, 13, 6, ..., A000593(k), ...];
n=1: [1, 2, 3, 4, 5, 6, 7, 8, 9, 10, ...];
n=2: [1, 3, 6, 10, 15, 21, 28, 36, 45, 55, ...];
n=3: [4, 4, 10, 26, 35, 56, 93, 120, 165, 232, ...];
n=4: [1, 5, 15, 35, 70, 126, 210, 330, 495, 715, ...];
n=5: [6, 6, 21, 56, 126, 262, 462, 792, 1287, 2002, ...];
n=6: [4, 7, 28, 93, 210, 462, 942, 1716, 3003, 5035, ...];
n=7: [8, 8, 36, 120, 330, 792, 1716, 3446, 6435, 11440, ...];
n=8: [1, 9, 45, 165, 495, 1287, 3003, 6435, 12870, 24310, ...];
n=9: [13, 10, 55, 232, 715, 2002, 5035, 11440, 24310, 48698, ...];
n=10: [6, 11, 66, 286, 1001, 3018, 8008, 19448, 43758, 92378, ...]; ...
in which the diagonal of coefficients of x^n*y^n/(2*n) equals
[0, 2, 6, 26, 70, 262, 942, 3446, 12870, 48698, ..., 2*a(n), ...],
which is twice this sequence.
The related infinite product may be written as the following series expansion
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) * (1 - x^11 - y^11) * ...) = 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) + ...

Links

Crossrefs

Cf. A322188, A322198, A322203, A000593.

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)) ); }
{a(n) = polcoeff( n*polcoeff( L, n, x), n, y)}
for(n=1, N, print1( a(n), ", ") )

Formula

a(n) ~ 2^(2*n-1) / sqrt(Pi*n). - Vaclav Kotesovec, Jun 18 2019

A322201 Main diagonal of square table A322200.

Original entry on oeis.org

0, 2, 10, 26, 90, 262, 994, 3446, 13050, 48698, 185310, 705454, 2706354, 10400626, 40123534, 155118406, 601106490, 2333606254, 9075235522, 35345263838, 137846899790, 538257884918, 2104100374694, 8233430727646, 32247609134418, 126410606439062, 495918553749434, 1946939425794206, 7648690681007998, 30067266499541098, 118264581875657214, 465428353255261150
Offset: 0

Views

Author

Paul D. Hanna, Nov 30 2018

Keywords

Examples

			L.g.f.: L(x) = 2*x + 10*x^2/2 + 26*x^3/3 + 90*x^4/4 + 262*x^5/5 + 994*x^6/6 + 3446*x^7/7 + 13050*x^8/8 + 48698*x^9/9 + 185310*x^10/10 + 705454*x^11/11 + 2706354*x^12/12 + ...
such that
exp( L(x) ) = 1 + 2*x + 7*x^2 + 20*x^3 + 63*x^4 + 190*x^5 + 613*x^6 + 1976*x^7 + 6604*x^8 + 22368*x^9 + 77270*x^10 + 270208*x^11 + 956780*x^12 + ...

Links

Crossrefs

Cf. A322200, A322202, A322203, A322205, A322207, A322209.

Programs

  • PARI
    {L = sum(n=1,61, -log(1 - (x^n + y^n) +O(x^61) +O(y^61)) );}
{a(n) = polcoeff( 2*n*polcoeff( L,n,x),n,y)}
for(n=0,35, print1( a(n),", ") )

Formula

a(n) = coefficient of x^n*y^n/(2*n) in Sum_{n>=1} -log(1 - (x^n + y^n)) for n>=0.
a(n) ~ 4^n / sqrt(Pi*n). - Vaclav Kotesovec, Jun 18 2019

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.

A322208 G.f.: exp( Sum_{n>=1} A322207(n)*x^n/n ), where A322207(n) is the coefficient of x^(3*n)*y^n/n in Sum_{n>=1} -log(1 - (x^n + y^n)).

Original entry on oeis.org

1, 1, 5, 24, 150, 1002, 7296, 55082, 429803, 3429141, 27861573, 229668027, 1916090676, 16147650896, 137259255191, 1175441115628, 10131538868330, 87826869133114, 765203002559216, 6697119583569563, 58852148074050440, 519073825025517314, 4593478958169093555, 40773010611894321971, 362920132925603812683, 3238611637275915021439, 28968760785263718554360
Offset: 0

Views

Author

Paul D. Hanna, Dec 01 2018

Keywords

Examples

			G.f.: A(x) = 1 + x + 5*x^2 + 24*x^3 + 150*x^4 + 1002*x^5 + 7296*x^6 + 55082*x^7 + 429803*x^8 + 3429141*x^9 + 27861573*x^10 + 229668027*x^11 + 1916090676*x^12 + ...
such that
log( A(x) ) = x + 9*x^2/2 + 58*x^3/3 + 473*x^4/4 + 3881*x^5/5 + 33786*x^6/6 + 296017*x^7/7 + 2630521*x^8/8 + 23535994*x^9/9 + 211922929*x^10/10 + ... + A322207(n)*x^n/n + ...
RELATED SERIES.
A(x)^4 = 1 + 4*x + 26*x^2 + 160*x^3 + 1099*x^4 + 7856*x^5 + 59090*x^6 + 457876*x^7 + 3639573*x^8 + 29479584*x^9 + 242474096*x^10 + ...

Crossrefs

Cf. A322200, A322207, A322205, A322203.

Programs

  • PARI
    {L = sum(n=1,121, -log(1 - (x^n + y^n) +O(x^121) +O(y^121)) );}
{A322207(n) = polcoeff( n*polcoeff( L,3*n,x),n,y)}
{a(n) = polcoeff( exp( sum(m=1,n, A322207(m)*x^m/m ) +x*O(x^n) ),n) }
for(n=0,40, print1( a(n),", ") )
Showing 1-6 of 6 results.

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