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 1-2 of 2 results.

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)).

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

Views

Author

Paul D. Hanna, Nov 30 2018

Keywords

Comments

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

Examples

			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 + ...

Links

Crossrefs

Cf. A322200, A322201, A322204.

Programs

  • Mathematica
    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 *)
  • PARI
    {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),", ") )

Formula

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

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).
Showing 1-2 of 2 results.

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