A322200 -id:A322200 - cp's OEIS frontend

A322201 Main diagonal of square table A322200.

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

Paul D. Hanna, Nov 30 2018

nonn

			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 + ...

Paul D. Hanna, Table of n, a(n) for n = 0..400

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

{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),", ") )

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

A322204 G.f.: exp( Sum_{n>=1} A322203(n)x^n/n ), where A322203(n) is the coefficient of x^ny^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

Paul D. Hanna, Nov 30 2018

nonn

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

			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 + ...

Alois P. Heinz, Table of n, a(n) for n = 0..1669 (first 401 terms from Paul D. Hanna)

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

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

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 *)

{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),", ") )

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

A322210 G.f.: P(x,y) = Product_{n>=1} 1/(1 - (x^n + y^n)), where P(x,y) = Sum_{n>=0} Sum_{k>=0} T(n,k) * x^n*y^k, as a square table of coefficients T(n,k) read by antidiagonals.

Original entry on oeis.org

1, 1, 1, 2, 2, 2, 3, 4, 4, 3, 5, 7, 10, 7, 5, 7, 12, 18, 18, 12, 7, 11, 19, 34, 38, 34, 19, 11, 15, 30, 56, 74, 74, 56, 30, 15, 22, 45, 94, 133, 158, 133, 94, 45, 22, 30, 67, 146, 233, 297, 297, 233, 146, 67, 30, 42, 97, 228, 385, 550, 602, 550, 385, 228, 97, 42, 56, 139, 340, 623, 951, 1166, 1166, 951, 623, 340, 139, 56

Offset: 0

Paul D. Hanna, Nov 30 2018

Conjecture 1: the triangular table T(n,k) is the number of ways to form the subsum k from the partitions of n, where n and k are integers such that 0 <= k <= n. For example, t(4,2)=10; the five partitions of 4 are (4), (3,1), (2,2), (2,1,1), (1,1,1,1) with subsum 2 occurring {0,0,2,2,6) times for a total of 10. - George Beck, Jan 03 2020
From Wouter Meeussen, Mar 09 2023: (Start)
Conjecture 2: the square table T(n,k) is the coefficient of s_lambda in the sum over all partitions lambda |-n and nu |-k of (s_rho/mu) where s_lambda*s_mu = Sum(rho|-n+k; C(rho, lambda, mu) s_rho). Simply stated as: multiply lambda with mu, and, for each term in the result, take the skew Schur function with mu and count how often you get the original lambda back. Sum up over all lambda and mu of the size n and k.
Conjecture 3: the triangular table T(n,k) is analogous to conjecture 2, but counting s_lambda in s_(lambda/mu) * s_mu with lambda |- n and mu |- k and 0<=k<=n. (End)

			G.f.: P(x,y) = 1 + (x + y) + (2*x^2 + 2*x*y + 2*y^2) + (3*x^3 + 4*x^2*y + 4*x*y^2 + 3*y^3) + (5*x^4 + 7*x^3*y + 10*x^2*y^2 + 7*x*y^3 + 5*y^4) + (7*x^5 + 12*x^4*y + 18*x^3*y^2 + 18*x^2*y^3 + 12*x*y^4 + 7*y^5) + (11*x^6 +19*x^5*y + 34*x^4*y^2 + 38*x^3*y^3 + 34*x^2*y^4 + 19*x*y^5 + 11*y^6) + (15*x^7 + 30*x^6*y + 56*x^5*y^2 + 74*x^4*y^3 + 74*x^3*y^4 + 56*x^2*y^5 + 30*x*y^6 + 15*y^7) + (22*x^8 + 45*x^7*y + 94*x^6*y^2 + 133*x^5*y^3 + 158*x^4*y^4 + 133*x^3*y^5 + 94*x^2*y^6 + 45*x*y^7 + 22*y^8) + ...
such that
P(x,y) = Product_{n>=1} 1/(1 - (x^n + y^n)),
where
P(x,y) = Sum_{n>=0} Sum_{k>=0} T(n,k) * x^n*y^k.
SQUARE TABLE.
The square table of coefficients T(n,k) of x^n*y^k in P(x,y) begins
   1,   1,   2,    3,    5,     7,    11,     15,     22,     30, ...
   1,   2,   4,    7,   12,    19,    30,     45,     67,     97, ...
   2,   4,  10,   18,   34,    56,    94,    146,    228,    340, ...
   3,   7,  18,   38,   74,   133,   233,    385,    623,    977, ...
   5,  12,  34,   74,  158,   297,   550,    951,   1614,   2627, ...
   7,  19,  56,  133,  297,   602,  1166,   2133,   3775,   6437, ...
  11,  30,  94,  233,  550,  1166,  2382,   4551,   8424,  14953, ...
  15,  45, 146,  385,  951,  2133,  4551,   9142,  17639,  32680, ...
  22,  67, 228,  623, 1614,  3775,  8424,  17639,  35492,  68356, ...
  30,  97, 340,  977, 2627,  6437, 14953,  32680,  68356, 136936, ...
  42, 139, 506, 1501, 4202, 10692, 25835,  58659, 127443, 264747, ...
  56, 195, 730, 2255, 6531, 17290, 43313, 102149, 229998, 495195, ...
  ...
TRIANGLE.
Alternatively, this sequence may be written as a triangle, starting as
   1;
   1,   1;
   2,   2,   2;
   3,   4,   4,   3;
   5,   7,  10,   7,    5;
   7,  12,  18,  18,   12,    7;
  11,  19,  34,  38,   34,   19,   11;
  15,  30,  56,  74,   74,   56,   30,   15;
  22,  45,  94, 133,  158,  133,   94,   45,   22;
  30,  67, 146, 233,  297,  297,  233,  146,   67,  30;
  42,  97, 228, 385,  550,  602,  550,  385,  228,  97,  42;
  56, 139, 340, 623,  951, 1166, 1166,  951,  623, 340, 139,  56;
  77, 195, 506, 977, 1614, 2133, 2382, 2133, 1614, 977, 506, 195, 77;
  ...

Alois P. Heinz, Antidiagonals n = 0..200 (first 61 antidiagonals from Paul D. Hanna)

Cf. A322200 (log).
Cf. A000041 (row 0 = partitions), A000070 (row 1), A093695(k+2) (row 2).
Main diagonal gives A322211.
Antidiagonal sums give A070933.
Cf. A284593.
Cf. A361286.

b:= proc(n, i) option remember; expand(`if`(n=0 or i=1,
      (x+1)^n, b(n, i-1) +(x^i+1)*b(n-i, min(n-i, i))))
    end:
T:= (n, k)-> coeff(b(n+k$2), x, k):
seq(seq(T(n, d-n), n=0..d), d=0..12);  # Alois P. Heinz, Aug 23 2019

b[n_, i_] := b[n, i] = Expand[If[n == 0 || i == 1, (x + 1)^n, b[n, i - 1] + (x^i + 1) b[n - i, Min[n - i, i]]]];
T[n_, k_] := Coefficient[b[n + k, n + k], x, k];
Table[Table[T[n, d - n], {n, 0, d}], {d, 0, 12}] // Flatten (* Jean-François Alcover, Dec 06 2019, after Alois P. Heinz *)

{P = 1/prod(n=1,61, (1 - (x^n + y^n) +O(x^61) +O(y^61)) );}
{T(n,k) = polcoeff( polcoeff( P,n,x),k,y)}
for(n=0,16, for(k=0,16, print1( T(n,k),", ") );print(""))

FORMULAS FOR TERMS.
T(n,k) = T(k,n) for n >= 0, k >= 0.
T(n,0) = A000041(n) for n >= 0, where A000041 is the partition numbers.
T(n,1) = A000070(n) for n >= 0, where A000070 is the sum of partitions.
ROW GENERATING FUNCTIONS.
Row 0: 1/( Product_{n>=1} (1 - x^n) ).
Row 1: 1/( (1-x) * Product_{n>=1} (1 - x^n) ).
Row 2: 2/( (1-x) * (1-x^2) * Product_{n>=1} (1 - x^n) ).

A322203 a(n) = coefficient of x^n*y^n/n in Sum_{n>=1} -log(1 - (x^n + y^n)), for n >= 1.

Original entry on oeis.org

1, 5, 13, 45, 131, 497, 1723, 6525, 24349, 92655, 352727, 1353177, 5200313, 20061767, 77559203, 300553245, 1166803127, 4537617761, 17672631919, 68923449895, 269128942459, 1052050187347, 4116715363823, 16123804567209, 63205303219531, 247959276874717, 973469712897103, 3824345340503999, 15033633249770549, 59132290937828607, 232714176627630575, 916312071072401757

Offset: 1

Paul D. Hanna, Nov 30 2018

nonn

			G.f.: L(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 + 352727*x^11/11 + 1353177*x^12/12 + ...
such that
exp( L(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 + ... + A322204(n)*x^n + ...

Paul D. Hanna, Table of n, a(n) for n = 1..400

Cf. A322204, A322202, A322200.

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

a(n) = A322200(n,n)/2 for n >= 1.

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

A054598 a(0)=0; for n>0, a(n) = Sum_{d|n} d*2^(n/d).

Original entry on oeis.org

0, 2, 8, 14, 32, 42, 104, 142, 320, 554, 1128, 2070, 4352, 8218, 16696, 32934, 66176, 131106, 263480, 524326, 1050912, 2097634, 4198488, 8388654, 16786688, 33554642, 67125352, 134219390, 268468960, 536870970, 1073811144, 2147483710, 4295099648, 8589940890

Offset: 0

N. J. A. Sloane, Apr 16 2000

nonn

Row sums of A322200, where A322200 describes Sum_{n>=1} -log(1 - (x^n + y^n)). - Paul D. Hanna, Dec 01 2018

Cf. A054599, A054600, A054601, A000016, A000031.

Cf. A322200.

Table[CoefficientList[Series[-Log[-QPochhammer[2, x]], {x, 0, 60}], x][[n]] (n - 1), {n, 1, 60}] (* Benedict W. J. Irwin, Jun 23 2016 *)

a(n) = sumdiv(n, d, d*2^(n/d)); \\ Michel Marcus, Jul 01 2016

L.g.f.: -log(Product_{ k>0 } (1-2*x^k)) = Sum_{ n>=0 } (a(n)/n)*x^n. - Benedict W. J. Irwin, Jun 23 2016

G.f.: Sum_{k>=1} 2^k*x^k/(1 - x^k)^2. - Ilya Gutkovskiy, Oct 24 2018

A322199 Expansion of Product_{k>=1} 1/(1 - (2^k + 1) * x^k).

Original entry on oeis.org

1, 3, 14, 51, 195, 663, 2345, 7707, 25744, 82980, 267812, 846150, 2676163, 8337189, 25947281, 80053128, 246468551, 754366239, 2305139065, 7014997404, 21317567297, 64606020012, 195557995054, 590855420007, 1783577678925, 5377112705874, 16199746640340, 48763788775530, 146712079122114, 441146762285301, 1326002750336702, 3984148679940612, 11967872331787643

Offset: 0

Paul D. Hanna, Dec 01 2018

nonn

This sequence is obtained from the generalized Euler transform in A266964 by taking f(n) = 1, g(n) = 2^n + 1. - Seiichi Manyama, Apr 11 2025

			G.f.: A(x) = 1 + 3*x + 14*x^2 + 51*x^3 + 195*x^4 + 663*x^5 + 2345*x^6 + 7707*x^7 + 25744*x^8 + 82980*x^9 + 267812*x^10 + 846150*x^11 + 2676163*x^12 + ...
such that
A(x) = 1/( (1 - 3*x) * (1 - 5*x^2) * (1 - 9*x^3) * (1 - 17*x^4) * (1 - 33*x^5) * (1 - 65*x^6) * (1 - 129*x^7) * ... * (1 - (2^n+1)*x^n) * ... ).
RELATED SERIES.
log( A(x) ) = 3*x + 19*x^2/2 + 54*x^3/3 + 199*x^4/4 + 408*x^5/5 + 1612*x^6/6 + 3090*x^7/7 + 11023*x^8/8 + 26487*x^9/9 + 80994*x^10/10 + 199686*x^11/11 + 676540*x^12/12 + ... + A322209(n)*x^n/n + ...

Vaclav Kotesovec, Table of n, a(n) for n = 0..1000
Seiichi Manyama, Generalized Euler transform.

Cf. A322209, A322200, A322210.

Cf. A000051, A266964.

{a(n) = polcoeff( 1/prod(m=1,n, 1 - (2^m+1)*x^m +x*O(x^n)),n)}
for(n=0,30, print1(a(n),", "))

G.f.: exp( Sum_{n>=1} x^n/n * Sum_{k=0..n} A322200(n-k,k) * 2^k ).
a(n) ~ c * 3^n, where c = Product_{k>=2} 1/(1 - (2^k + 1)/3^k) = 6.49344992975096517443610066284481821741772051973643441550853873760083... - Vaclav Kotesovec, Oct 04 2020
a(n) = Sum_{k=0..n} 2^k * A322210(k,n-k). - Seiichi Manyama, Apr 11 2025

A322202 G.f.: exp( Sum_{n>=1} A322201(n)x^n/n ), where A322201(n) is the coefficient of x^ny^n/(2*n) in Sum_{n>=1} -log(1 - (x^n + y^n)).

Original entry on oeis.org

1, 2, 7, 20, 63, 190, 613, 1976, 6604, 22368, 77270, 270208, 956780, 3419212, 12323226, 44723840, 163320766, 599601984, 2211844684, 8193734760, 30469278673, 113692852342, 425558528235, 1597428832560, 6011972255226, 22680620270712, 85754229105470, 324898592591960, 1233299357981416, 4689870496585016, 17863799895741982, 68149300647823612, 260364494604701847, 996086232267182566, 3815683108118138847, 14634441964549504036

Offset: 0

Paul D. Hanna, Nov 30 2018

nonn

Conjecture: Euler transform of A123611. - Vaclav Kotesovec, Dec 12 2020

			G.f.: A(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 + ...
such that
log( A(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 + ... + A322201(n)*x^n/n + ...
sqrt(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 + ... + A322204(n)*x^n + ...

Paul D. Hanna, Table of n, a(n) for n = 0..400

Cf. A322200, A322201, A322204.

nmax = 30; CoefficientList[Series[Product[Sum[CatalanNumber[k]*x^(j*k), {k, 0, nmax/j}]^2, {j, 1, nmax}], {x, 0, nmax}], x] (* Vaclav Kotesovec, Dec 12 2020 *)
nmax = 30; CoefficientList[Series[Product[(1 - Sqrt[1 - 4*x^k])/(2*x^k), {k, 1, nmax}]^2, {x, 0, nmax}], x] (* Vaclav Kotesovec, Dec 12 2020 *)

{L = sum(n=1,61, -log(1 - (x^n + y^n) +O(x^61) +O(y^61)) );}
{A322201(n) = polcoeff( 2*n*polcoeff( L,n,x),n,y)}
{a(n) = polcoeff( exp( sum(m=1,n, A322201(m)*x^m/m ) +x*O(x^n) ),n) }
for(n=0,35, print1( a(n),", ") )

a(n) ~ c * 4^n / n^(3/2), where c = 4/sqrt(Pi) * Product_{j>=1} (2^(j+1) * (2^j - sqrt(4^j - 1)))^2 = 2.704933139869066452954644773467... - Vaclav Kotesovec, Jun 18 2019, updated Dec 12 2020

G.f.: Product_{j>=1} c(x^j)^2, where c(x) = (1-sqrt(1-4*x))/(2*x) is the g.f. of A000108. - Vaclav Kotesovec, Dec 12 2020

A322205 a(n) = coefficient of x^(2n)y^n/n in Sum_{n>=1} -log(1 - (x^n + y^n)) for n >= 1.

Original entry on oeis.org

1, 7, 31, 179, 1006, 6265, 38767, 245515, 1562368, 10017042, 64512251, 417238925, 2707475161, 17620153929, 114955811686, 751616795579, 4923689695592, 32308786002880, 212327989773919, 1397281521970074, 9206478467570842, 60727722789611357, 400978991944396343, 2650087221531556021, 17529515713716302906, 116043807648704288815, 768759815833955021344, 5096278545391603271517

Offset: 1

Paul D. Hanna, Dec 01 2018

nonn

			G.f.: L(x) = x + 7*x^2/2 + 31*x^3/3 + 179*x^4/4 + 1006*x^5/5 + 6265*x^6/6 + 38767*x^7/7 + 245515*x^8/8 + 1562368*x^9/9 + 10017042*x^10/10 + 64512251*x^11/11 + 417238925*x^12/12 + ...
such that
exp( L(x) ) = 1 + x + 4*x^2 + 14*x^3 + 63*x^4 + 294*x^5 + 1526*x^6 + 8157*x^7 + 45332*x^8 + 257378*x^9 + 1489539*x^10 + 8744722*x^11 + 51965701*x^12 + ... + A322206(n)*x^n + ...

Cf. A322200, A322206.

{L = sum(n=1,81, -log(1 - (x^n + y^n) +O(x^81) +O(y^81)) );}
{a(n) = polcoeff( n*polcoeff( L,2*n,x),n,y)}
for(n=1,35, print1( a(n),", ") )

a(n) = A322200(2*n,n)/3.

A322207 a(n) = coefficient of x^(3n)y^n/n in Sum_{n>=1} -log(1 - (x^n + y^n)).

Original entry on oeis.org

1, 9, 58, 473, 3881, 33786, 296017, 2630521, 23535994, 211922929, 1917334794, 17417202554, 158753389913, 1451183583033, 13298522310098, 122131739530937, 1123787895356429, 10358022488568858, 95615237915961119, 883829035976891713, 8179808679273553156, 75788358479315971850, 702916267465270526873, 6525429588311530420858, 60629817430084280273281

Offset: 1

Paul D. Hanna, Dec 01 2018

nonn

			G.f.: L(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 + 1917334794*x^11/11 + 17417202554*x^12/12 + ...
such that
exp( L(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 + ... + A322208(n)*x^n + ...

Cf. A322200, A322208, A322206, A322204.

{L = sum(n=1,121, -log(1 - (x^n + y^n) +O(x^121) +O(y^121)) );}
{a(n) = polcoeff( n*polcoeff( L,3*n,x),n,y)}
for(n=1,40, print1( a(n),", ") )

a(n) = A322200(3*n,n)/4.

Logarithmic derivative of A322208.

A322209 L.g.f.: log( Product_{n>=1} 1/(1 - (2^n+1)*x^n) ).

Original entry on oeis.org

0, 3, 19, 54, 199, 408, 1612, 3090, 11023, 26487, 80994, 199686, 676540, 1700832, 5285096, 15197274, 45739039, 131368404, 401655943, 1172222958, 3549402474, 10533769146, 31617172980, 94336116834, 283990486780, 848323147233, 2546924693306, 7631598676410, 22903854049016, 68645946621360, 206035134959112, 617739968277066, 1853594327953471

Offset: 0

Paul D. Hanna, Dec 01 2018

nonn

			L.g.f.: L(x) = 3*x + 19*x^2/2 + 54*x^3/3 + 199*x^4/4 + 408*x^5/5 + 1612*x^6/6 + 3090*x^7/7 + 11023*x^8/8 + 26487*x^9/9 + 80994*x^10/10 + 199686*x^11/11 + 676540*x^12/12 + ...
such that
exp( L(x) ) = 1 + 3*x + 14*x^2 + 51*x^3 + 195*x^4 + 663*x^5 + 2345*x^6 + 7707*x^7 + 25744*x^8 + 82980*x^9 + 267812*x^10 + 846150*x^11 + 2676163*x^12 + ... + A322199(n)*x^n + ...
also,
exp( L(x) ) = 1/( (1 - 3*x) * (1 - 5*x^2) * (1 - 9*x^3) * (1 - 17*x^4) * (1 - 33*x^5) * (1 - 65*x^6) * (1 - 129*x^7) * ... * (1 - (2^n+1)*x^n) * ... ).

Cf. A322200, A322199.

{L = sum(n=1,41, -log(1 - (x^n + y^n) +O(x^41) +O(y^41)) );}
{A322200(n,k) = polcoeff( (n+k)*polcoeff( L,n,x),k,y)}
{a(n) = sum(k=0,n, A322200(n-k,k)*2^k )}
for(n=0,40, print1( a(n),", ") )

a(n) = Sum_{k=0..n} A322200(n-k,k) * 2^k for n >= 0.

cp's OEIS Frontend

A322201 Main diagonal of square table A322200.

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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)).

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A322210 G.f.: P(x,y) = Product_{n>=1} 1/(1 - (x^n + y^n)), where P(x,y) = Sum_{n>=0} Sum_{k>=0} T(n,k) * x^n*y^k, as a square table of coefficients T(n,k) read by antidiagonals.

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A322203 a(n) = coefficient of x^n*y^n/n in Sum_{n>=1} -log(1 - (x^n + y^n)), for n >= 1.

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A054598 a(0)=0; for n>0, a(n) = Sum_{d|n} d*2^(n/d).

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A322199 Expansion of Product_{k>=1} 1/(1 - (2^k + 1) * x^k).

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A322202 G.f.: exp( Sum_{n>=1} A322201(n)*x^n/n ), where A322201(n) is the coefficient of x^n*y^n/(2*n) in Sum_{n>=1} -log(1 - (x^n + y^n)).

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

A322202 G.f.: exp( Sum_{n>=1} A322201(n)x^n/n ), where A322201(n) is the coefficient of x^ny^n/(2*n) in Sum_{n>=1} -log(1 - (x^n + y^n)).

A322205 a(n) = coefficient of x^(2n)y^n/n in Sum_{n>=1} -log(1 - (x^n + y^n)) for n >= 1.

A322207 a(n) = coefficient of x^(3n)y^n/n in Sum_{n>=1} -log(1 - (x^n + y^n)).