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 1-5 of 5 results.

A354650 G.f. A(x,y) satisfies: -y = f(-x,-A(x,y)), where f(x,y) = Sum_{n=-oo..oo} x^(n*(n+1)/2) * y^(n*(n-1)/2) is Ramanujan's theta function.

Original entry on oeis.org

1, 1, 0, 3, 3, 1, 0, 9, 27, 30, 15, 3, 0, 22, 147, 340, 390, 246, 83, 12, 0, 51, 630, 2530, 5070, 5928, 4284, 1908, 486, 55, 0, 108, 2295, 14595, 45450, 83559, 98910, 78282, 41580, 14355, 2937, 273, 0, 221, 7476, 70737, 319605, 849450, 1472261, 1757688, 1484451, 891890, 375442, 105930, 18109, 1428, 0, 429, 22302, 301070, 1886010, 6878907, 16386636, 27205308, 32683680, 28981855, 19081854, 9258678, 3231514, 771225, 113220, 7752
Offset: 0

Views

Author

Paul D. Hanna, Jun 02 2022

Keywords

Comments

Unsigned version of A354649.
Column 1 equals A000716, with g.f. P(x)^3 where P(x) = exp( Sum_{n>=1} sigma(n)*x^n/n ) is the partition function.
The rightmost border equals A001764, with g.f. C(x) = 1 + x*C(x)^3.
T(n,1) = A000716(n), for n >= 0.
T(n,2) = A354655(n), for n >= 1.
T(n,3) = A354656(n), for n >= 1.
T(n,n) = A354658(n), for n >= 0.
T(n,n+1) = A354659(n), for n >= 0.
T(n,2*n) = A354660(n), for n >= 0.
T(n,2*n+1) = A001764(n), for n >= 0.
Antidiagonal sums = A268650.
Row sums = A268299 (with offset).
Sum_{k=0..2*n+1} T(n,k)*2^k = A354652(n), for n >= 0.
Sum_{k=0..2*n+1} T(n,k)*3^k = A354653(n), for n >= 0.
Sum_{k=0..2*n+1} T(n,k)*4^k = A354654(n), for n >= 0.
Sum_{k=0..2*n+1} T(n,k)*(-1)^k = -A354661(n), for n >= 0.
Sum_{k=0..2*n+1} T(n,k)*(-2)^k = -A354662(n), for n >= 0.
Sum_{k=0..2*n+1} T(n,k)*(-3)^k = -A354663(n), for n >= 0.
Sum_{k=0..2*n+1} T(n,k)*(-4)^k = -A354664(n), for n >= 0.
SPECIFIC VALUES.
(1) A(x,y) = -exp(-Pi) at x = -exp(-Pi), y = -Pi^(1/4)/gamma(3/4).
(2) A(x,y) = -exp(-2*Pi) at x = -exp(-2*Pi), y = -Pi^(1/4)/gamma(3/4) * (6 + 4*sqrt(2))^(1/4)/2.
(3) A(x,y) = -exp(-3*Pi) at x = -exp(-3*Pi), y = -Pi^(1/4)/gamma(3/4) * (27 + 18*sqrt(3))^(1/4)/3.
(4) A(x,y) = -exp(-4*Pi) at x = -exp(-4*Pi), y = -Pi^(1/4)/gamma(3/4) * (8^(1/4) + 2)/4.
(5) A(x,y) = -exp(-sqrt(3)*Pi) at x = -exp(-sqrt(3)*Pi), y = -gamma(4/3)^(3/2)*3^(13/8)/(Pi*2^(2/3)).

Examples

			G.f.: A(x,y) = (1 + y) + x*(3*y + 3*y^2 + y^3) + x^2*(9*y + 27*y^2 + 30*y^3 + 15*y^4 + 3*y^5) + x^3*(22*y + 147*y^2 + 340*y^3 + 390*y^4 + 246*y^5 + 83*y^6 + 12*y^7) + x^4*(51*y + 630*y^2 + 2530*y^3 + 5070*y^4 + 5928*y^5 + 4284*y^6 + 1908*y^7 + 486*y^8 + 55*y^9) + x^5*(108*y + 2295*y^2 + 14595*y^3 + 45450*y^4 + 83559*y^5 + 98910*y^6 + 78282*y^7 + 41580*y^8 + 14355*y^9 + 2937*y^10 + 273*y^11) + ...
such that A = A(x,y) satisfies:
(1) -y = ... + x^36*A^28 - x^28*A^21 + x^21*A^15 - x^15*A^10 + x^10*A^6 - x^6*A^3 + x^3*A - x + 1 - A + x*A^3 - x^3*A^6 + x^6*A^10 - x^10*A^15 + x^15*A^21 - x^21*A^28 + x^28*A^36 + ...
(2) -y = (1 - x*A)*(1 - A)*(1-x) * (1 - x^2*A^2)*(1 - x*A^2)*(1 - x^2*A) * (1 - x^3*A^3)*(1 - x^2*A^3)*(1 - x^3*A^2) * (1 - x^4*A^4)*(1 - x^3*A^4)*(1 - x^4*A^3) * (1 - x^5*A^5)*(1 - x^4*A^5)*(1 - x^5*A^4) * ...
(3) -y = (1-x) - (1-x^3)*A + x*(1-x^5)*A^3 - x^3*(1-x^7)*A^6 + x^6*(1-x^9)*A^10 - x^10*(1-x^11)*A^15 + x^15*(1-x^13)*A^21 - x^21*(1-x^15)*A^28 + ...
(4) -y = (1-A) - (1-A^3)*x + A*(1-A^5)*x^3 - A^3*(1-A^7)*x^6 + A^6*(1-A^9)*x^10 - A^10*(1-A^11)*x^15 + A^15*(1-A^13)*x^21 - A^21*(1-A^15)*x^28 + ...
This triangle of coefficients of x^n*y^k in g.f. A(x,y) for n >= 0, k = 0..2*n+1, begins:
1, 1;
0, 3, 3, 1;
0, 9, 27, 30, 15, 3;
0, 22, 147, 340, 390, 246, 83, 12;
0, 51, 630, 2530, 5070, 5928, 4284, 1908, 486, 55;
0, 108, 2295, 14595, 45450, 83559, 98910, 78282, 41580, 14355, 2937, 273;
0, 221, 7476, 70737, 319605, 849450, 1472261, 1757688, 1484451, 891890, 375442, 105930, 18109, 1428;
0, 429, 22302, 301070, 1886010, 6878907, 16386636, 27205308, 32683680, 28981855, 19081854, 9258678, 3231514, 771225, 113220, 7752;
0, 810, 62100, 1157820, 9729720, 46977378, 147584556, 324283068, 520974180, 628884300, 579226362, 409367712, 221218179, 90115620, 26879160, 5559408, 715122, 43263; ...
The rightmost border equals A001764, with g.f. C(x) = 1 + x*C(x)^3.
Column 1 equals A000716, with g.f. P(x)^3 where P(x) = exp( Sum_{n>=1} x^n/(n*(1-x^n)) ) is the partition function.

Links

Crossrefs

Cf. A000716 (column 1), A354655 (column 2), A354656 (column 3).
Cf. A354658 (T(n,n)), A354659 (T(n,n+1)), A354660 (T(n,2*n)), A001764 (right border).
Cf. A268299 (y=1), A354652 (y=2), A354653 (y=3), A354654 (y=4).
Cf. A354661 (y=-1), A354662 (y=-2), A354663 (y=-3), A354664 (y=-4).
Cf. A268650 (antidiagonal sums), A354657, A354649.

Programs

  • PARI
    {T(n,k) = my(A=[1+y]); for(i=1,n, A = concat(A,0);
A[#A] = polcoeff(y + sum(m=0,sqrtint(2*#A+9), (-1)^m * x^(m*(m-1)/2) * (1 - x^(2*m+1)) * Ser(A)^(m*(m+1)/2) ),#A-1) );
polcoeff(A[n+1],k,y)}
for(n=0,12,for(k=0,2*n+1,print1(T(n,k),", "));print(""))

Formula

G.f. A(x,y) = Sum_{n>=0} x^n * Sum_{k=0..2*n+1} T(n,k)*y^k satisfies:
(1) -y = A(-x,-f(x,y)) = Sum_{n>=0} (-x)^n * Sum_{k=0..2*n+1} (-1)^n * T(n,k) * f(x,y)^k, where f(,) is Ramanujan's theta function.
(2) -y = f(-x,-A(x,y)) = Sum_{n=-oo..oo} (-1)^n * x^(n*(n-1)/2) * A(x,y)^(n*(n+1)/2), where f(,) is Ramanujan's theta function.
(3) -y = Product_{n>=1} (1 - x^n*A(x,y)^n) * (1 - x^(n-1)*A(x,y)^n) * (1 - x^n*A(x,y)^(n-1)), by the Jacobi triple product identity.
(4) -y = Sum_{n>=0} (-1)^n * x^(n*(n-1)/2) * (1 - x^(2*n+1)) * A(x,y)^(n*(n+1)/2).
(5) -y = Sum_{n>=0} (-1)^n * A(x,y)^(n*(n-1)/2) * (1 - A(x,y)^(2*n+1)) * x^(n*(n+1)/2).
Formulas for terms in rows.
(6) T(n,1) = A000716(n), the number of partitions of n into parts of 3 kinds.
(7) T(n,2*n+1) = A001764(n) = binomial(3*n,n)/(2*n+1), for n >= 0.

A354649 G.f. A(x,y) satisfies: y = f(x,A(x,y)), where f(x,y) = Sum_{n=-oo..oo} x^(n*(n+1)/2) * y^(n*(n-1)/2) is Ramanujan's theta function.

Original entry on oeis.org

-1, 1, 0, -3, 3, -1, 0, 9, -27, 30, -15, 3, 0, -22, 147, -340, 390, -246, 83, -12, 0, 51, -630, 2530, -5070, 5928, -4284, 1908, -486, 55, 0, -108, 2295, -14595, 45450, -83559, 98910, -78282, 41580, -14355, 2937, -273, 0, 221, -7476, 70737, -319605, 849450, -1472261, 1757688, -1484451, 891890, -375442, 105930, -18109, 1428, 0, -429, 22302, -301070, 1886010, -6878907, 16386636, -27205308, 32683680, -28981855, 19081854, -9258678, 3231514, -771225, 113220, -7752
Offset: 0

Views

Author

Paul D. Hanna, Jun 02 2022

Keywords

Comments

Signed version of A354650.
Column 1 equals signed A000716, with g.f. P(-x)^3 where P(x) = exp( Sum_{n>=1} sigma(n)*x^n/n ) is the partition function.
The rightmost border equals signed A001764, with g.f. C(x) = 1 - x*C(x)^3.
T(n,1) = (-1)^n * A000716(n), for n >= 0.
T(n,2) = (-1)^(n+1) * A354655(n), for n >= 1.
T(n,3) = (-1)^n * A354656(n), for n >= 1.
T(n,n) = -A354658(n), for n >= 0.
T(n,n+1) = A354659(n), for n >= 0.
T(n,2*n) = (-1)^(n+1) * A354660(n), for n >= 0.
T(n,2*n+1) = (-1)^n * A001764(n), for n >= 0.
Antidiagonal sums equals signed A268650.
Sum_{k=0..2*n+1} T(n,k)*(-1)^k = (-1)^(n+1) * A268299(n+1), for n >= 0.
Sum_{k=0..2*n+1} T(n,k)*(-2)^k = (-1)^(n+1) * A354652(n), for n >= 0.
Sum_{k=0..2*n+1} T(n,k)*(-3)^k = (-1)^(n+1) * A354653(n), for n >= 0.
Sum_{k=0..2*n+1} T(n,k)*(-4)^k = (-1)^(n+1) * A354654(n), for n >= 0.
Sum_{k=0..2*n+1} T(n,k) = (-1)^n * A354661(n), for n >= 0.
Sum_{k=0..2*n+1} T(n,k)*2^k = (-1)^n * A354662(n), for n >= 0.
Sum_{k=0..2*n+1} T(n,k)*3^k = (-1)^n * A354663(n), for n >= 0.
Sum_{k=0..2*n+1} T(n,k)*4^k = (-1)^n * A354664(n), for n >= 0.
SPECIFIC VALUES.
(1) A(x,y) = exp(-Pi) at x = exp(-Pi), y = Pi^(1/4)/gamma(3/4).
(2) A(x,y) = exp(-2*Pi) at x = exp(-2*Pi), y = Pi^(1/4)/gamma(3/4) * (6 + 4*sqrt(2))^(1/4)/2.
(3) A(x,y) = exp(-3*Pi) at x = exp(-3*Pi), y = Pi^(1/4)/gamma(3/4) * (27 + 18*sqrt(3))^(1/4)/3.
(4) A(x,y) = exp(-4*Pi) at x = exp(-4*Pi), y = Pi^(1/4)/gamma(3/4) * (8^(1/4) + 2)/4.
(5) A(x,y) = exp(-sqrt(3)*Pi) at x = exp(-sqrt(3)*Pi), y = gamma(4/3)^(3/2)*3^(13/8)/(Pi*2^(2/3)).

Examples

			G.f.: A(x,y) = (-1 + y) - x*(3*y - 3*y^2 + y^3) + x^2*(9*y - 27*y^2 + 30*y^3 - 15*y^4 + 3*y^5) - x^3*(22*y - 147*y^2 + 340*y^3 - 390*y^4 + 246*y^5 - 83*y^6 + 12*y^7) + x^4*(51*y - 630*y^2 + 2530*y^3 - 5070*y^4 + 5928*y^5 - 4284*y^6 + 1908*y^7 - 486*y^8 + 55*y^9) - x^5*(108*y - 2295*y^2 + 14595*y^3 - 45450*y^4 + 83559*y^5 - 98910*y^6 + 78282*y^7 - 41580*y^8 + 14355*y^9 - 2937*y^10 + 273*y^11) + x^6*(221*y - 7476*y^2 + 70737*y^3 - 319605*y^4 + 849450*y^5 - 1472261*y^6 + 1757688*y^7 - 1484451*y^8 + 891890*y^9 - 375442*y^10 + 105930*y^11 - 18109*y^12 + 1428*y^13) + x^7*(-429*y + 22302*y^2 - 301070*y^3 + 1886010*y^4 - 6878907*y^5 + 16386636*y^6 - 27205308*y^7 + 32683680*y^8 - 28981855*y^9 + 19081854*y^10 - 9258678*y^11 + 3231514*y^12 - 771225*y^13 + 113220*y^14 - 7752*y^15) + x^8*(810*y - 62100*y^2 + 1157820*y^3 - 9729720*y^4 + 46977378*y^5 - 147584556*y^6 + 324283068*y^7 - 520974180*y^8 + 628884300*y^9 - 579226362*y^10 + 409367712*y^11 - 221218179*y^12 + 90115620*y^13 - 26879160*y^14 + 5559408*y^15 - 715122*y^16 + 43263*y^17) + ...
such that A = A(x,y) satisfies:
(1) y = ... + x^36*A^28 + x^28*A^21 + x^21*A^15 + x^15*A^10 + x^10*A^6 + x^6*A^3 + x^3*A + x + 1 + A + x*A^3 + x^3*A^6 + x^6*A^10 + x^10*A^15 + x^15*A^21 + x^21*A^28 + x^28*A^36 + ...
(2) y = (1 - x*A)*(1 + A)*(1+x) * (1 - x^2*A^2)*(1 + x*A^2)*(1 + x^2*A) * (1 - x^3*A^3)*(1 + x^2*A^3)*(1 + x^3*A^2) * (1 - x^4*A^4)*(1 + x^3*A^4)*(1 + x^4*A^3) * (1 - x^5*A^5)*(1 + x^4*A^5)*(1 + x^5*A^4) * ...
(3) y = (1+x) + (1+x^3)*A + x*(1+x^5)*A^3 + x^3*(1+x^7)*A^6 + x^6*(1+x^9)*A^10 + x^10*(1+x^11)*A^15 + x^15*(1+x^13)*A^21 + x^21*(1+x^15)*A^28 + ...
(4) y = (1+A) + (1+A^3)*x + A*(1+A^5)*x^3 + A^3*(1+A^7)*x^6 + A^6*(1+A^9)*x^10 + A^10*(1+A^11)*x^15 + A^15*(1+A^13)*x^21 + A^21*(1+A^15)*x^28 + ...
This triangle of coefficients of x^n*y^k in g.f. A(x,y) for n >= 0, k = 0..2*n+1, begins:
-1, 1;
0, -3, 3, -1;
0, 9, -27, 30, -15, 3;
0, -22, 147, -340, 390, -246, 83, -12;
0, 51, -630, 2530, -5070, 5928, -4284, 1908, -486, 55;
0, -108, 2295, -14595, 45450, -83559, 98910, -78282, 41580, -14355, 2937, -273;
0, 221, -7476, 70737, -319605, 849450, -1472261, 1757688, -1484451, 891890, -375442, 105930, -18109, 1428;
0, -429, 22302, -301070, 1886010, -6878907, 16386636, -27205308, 32683680, -28981855, 19081854, -9258678, 3231514, -771225, 113220, -7752;
0, 810, -62100, 1157820, -9729720, 46977378, -147584556, 324283068, -520974180, 628884300, -579226362, 409367712, -221218179, 90115620, -26879160, 5559408, -715122, 43263; ...
The rightmost border equals signed A001764, with g.f. C(x) = 1 - x*C(x)^3.
Column 1 equals signed A000716, with g.f. P(-x)^3 where P(x) = exp( Sum_{n>=1} x^n/(n*(1-x^n)) ) is the partition function.

Links

Crossrefs

Cf. A000716 (column 1), A354655 (column 2), A354656 (column 3).
Cf. A354658 (T(n,n)), A354659 (T(n,n+1)), A354660 (T(n,2*n)), A001764 (right border).
Cf. A268299 (y=-1), A354652 (y=-2), A354653 (y=-3), A354654 (y=-4).
Cf. A354661 (y=1), A354662 (y=2), A354663 (y=3), A354664 (y=4).
Cf. A268650 (antidiagonal sums), A354657, A354650.

Programs

  • PARI
    {T(n,k) = my(A=[y-1]); for(i=1,n, A = concat(A,0);
A[#A] = polcoeff(y - sum(m=0,sqrtint(2*#A+9), x^(m*(m-1)/2) * (1 + x^(2*m+1)) * Ser(A)^(m*(m+1)/2) ),#A-1) );
H=A; polcoeff(A[n+1],k,y)}
for(n=0,12,for(k=0,2*n+1,print1(T(n,k),", "));print(""))

Formula

G.f. A(x,y) = Sum_{n>=0} x^n * Sum_{k=0..2*n+1} T(n,k)*y^k satisfies:
(1) y = A(x,f(x,y)) = Sum_{n>=0} x^n * Sum_{k=0..2*n+1} T(n,k) * f(x,y)^k, where f(,) is Ramanujan's theta function.
(2) y = f(x,A(x,y)) = Sum_{n=-oo..oo} x^(n*(n-1)/2) * A(x,y)^(n*(n+1)/2), where f(,) is Ramanujan's theta function.
(3) y = Product_{n>=1} (1 - x^n*A(x,y)^n) * (1 + x^(n-1)*A(x,y)^n) * (1 + x^n*A(x,y)^(n-1)), by the Jacobi triple product identity.
(4) y = Sum_{n>=0} x^(n*(n-1)/2) * (1 + x^(2*n+1)) * A(x,y)^(n*(n+1)/2).
(5) y = Sum_{n>=0} A(x,y)^(n*(n-1)/2) * (1 + A(x,y)^(2*n+1)) * x^(n*(n+1)/2).
(6) T(n,1) = (-1)^n * A000716(n), where A000716(n) is the number of partitions of n into parts of 3 kinds.
(7) T(n,2*n+1) = (-1)^n * A001764(n) = (-1)^n * binomial(3*n,n)/(2*n+1), for n >= 0.

A355350 G.f. A(x,y) satisfies: x*y = Sum_{n=-oo..+oo} (-1)^n * x^(n*(n+1)/2) * A(x,y)^n, with coefficients T(n,k) of x^n*y^k in A(x,y) given as a triangle read by rows.

Original entry on oeis.org

1, 0, 1, 0, 3, 1, 0, 9, 6, 1, 0, 22, 27, 10, 1, 0, 51, 98, 66, 15, 1, 0, 108, 315, 340, 135, 21, 1, 0, 221, 918, 1495, 910, 246, 28, 1, 0, 429, 2492, 5838, 5070, 2086, 413, 36, 1, 0, 810, 6372, 20805, 24543, 14280, 4284, 652, 45, 1, 0, 1479, 15525, 68816, 106535, 83559, 35168, 8100, 981, 55, 1, 0, 2640, 36280, 213945, 423390, 432930, 243208, 78282, 14355, 1420, 66, 1
Offset: 0

Views

Author

Paul D. Hanna, Jun 29 2022

Keywords

Comments

The term T(n,k) is found in row n and column k of this triangle, and can be used to derive the following sequences.
A355351(n) = Sum_{k=0..n} T(n,k) for n >= 0 (row sums).
A355352(n) = Sum_{k=0..n} T(n,k) * 2^k for n >= 0.
A355353(n) = Sum_{k=0..n} T(n,k) * 3^k for n >= 0.
A355354(n) = Sum_{k=0..n} T(n,k) * 4^k for n >= 0.
A355355(n) = Sum_{k=0..n} T(n,k) * 5^k for n >= 0.
A355356(n) = Sum_{k=0..floor(n/2)} T(n-k,k) for n >= 0 (antidiagonal sums).
A355357(n) = Sum_{k=0..floor(n/2)} T(n-k,n-2*k) for n >= 0.
A354658(n) = T(2*n,n) for n >= 0 (central terms of this triangle).
Conjectures:
(C.1) Column 1 equals A000716, the number of partitions into parts of 3 kinds;
(C.2) Column 2 equals A023005, the number of partitions into parts of 6 kinds.

Examples

			G.f.: A(x,y) = 1 + x*y + x^2*(3*y + y^2) + x^3*(9*y + 6*y^2 + y^3) + x^4*(22*y + 27*y^2 + 10*y^3 + y^4) + x^5*(51*y + 98*y^2 + 66*y^3 + 15*y^4 + y^5) + x^6*(108*y + 315*y^2 + 340*y^3 + 135*y^4 + 21*y^5 + y^6) + x^7*(221*y + 918*y^2 + 1495*y^3 + 910*y^4 + 246*y^5 + 28*y^6 + y^7) + x^8*(429*y + 2492*y^2 + 5838*y^3 + 5070*y^4 + 2086*y^5 + 413*y^6 + 36*y^7 + y^8) + x^9*(810*y + 6372*y^2 + 20805*y^3 + 24543*y^4 + 14280*y^5 + 4284*y^6 + 652*y^7 + 45*y^8 + y^9) + x^10*(1479*y + 15525*y^2 + 68816*y^3 + 106535*y^4 + 83559*y^5 + 35168*y^6 + 8100*y^7 + 981*y^8 + 55*y^9 + y^10) + ...
where
x*y = ... - x^10/A(x,y)^5 + x^6/A(x,y)^4 - x^3/A(x,y)^3 + x/A(x,y)^2 - 1/A(x,y) + 1 - x*A(x,y) + x^3*A(x,y)^2 - x^6*A(x,y)^3 + x^10*A(x,y)^4 -+ ... + (-1)^n * x^(n*(n+1)/2) * A(x,y)^n + ...
also, given P(x) is the partition function (A000041),
x*y*P(x) = (1 - x*A(x,y))*(1 - 1/A(x,y)) * (1 - x^2*A(x,y))*(1 - x/A(x,y)) * (1 - x^3*A(x,y))*(1 - x^2/A(x,y)) * (1 - x^4*A(x,y))*(1 - x^3/A(x,y)) * ... * (1 - x^n*A(x,y))*(1 - x^(n-1)/A(x,y)) * ...
TRIANGLE.
The triangle of coefficients T(n,k) of x^n*y^k in A(x,y), for k = 0..n in row n, begins:
n=0: [1];
n=1: [0, 1];
n=2: [0, 3, 1];
n=3: [0, 9, 6, 1];
n=4: [0, 22, 27, 10, 1];
n=5: [0, 51, 98, 66, 15, 1];
n=6: [0, 108, 315, 340, 135, 21, 1];
n=7: [0, 221, 918, 1495, 910, 246, 28, 1];
n=8: [0, 429, 2492, 5838, 5070, 2086, 413, 36, 1];
n=9: [0, 810, 6372, 20805, 24543, 14280, 4284, 652, 45, 1];
n=10: [0, 1479, 15525, 68816, 106535, 83559, 35168, 8100, 981, 55, 1];
n=11: [0, 2640, 36280, 213945, 423390, 432930, 243208, 78282, 14355, 1420, 66, 1];
n=12: [0, 4599, 81816, 630890, 1563705, 2033244, 1472261, 629280, 160965, 24145, 1991, 78, 1];
...
in which column 1 appears to equal A000716, the coefficients in P(x)^3,
and column 2 appears to equal A023005, the coefficients in P(x)^6,
where P(x) is the partition function and begins
P(x) = 1 + x + 2*x^2 + 3*x^3 + 5*x^4 + 7*x^5 + 11*x^6 + 15*x^7 + 22*x^8 + 30*x^9 + 42*x^10 + ... + A000041(n)*x^n + ...
Also, the power series expansions of P(x)^3 and P(x)^6 begin
P(x)^3 = 1 + 3*x + 9*x^2 + 22*x^3 + 51*x^4 + 108*x^5 + 221*x^6 + 429*x^7 + 810*x^8 + 1479*x^9 + 2640*x^10 + ... + A000716(n)*x^n + ...
P(x)^6 = 1 + 6*x + 27*x^2 + 98*x^3 + 315*x^4 + 918*x^5 + 2492*x^6 + 6372*x^7 + 15525*x^8 + 36280*x^9 + 81816*x^10 + ... + A023005(n)*x^n + ...

Links

Crossrefs

Cf. A355351 (row sums), A355352, A355353, A355354, A355355.
Cf. A355356, A355357, A354658 (central terms).
Cf. A354645, A354650 (related table), A000041, A000716, A023005.

Programs

  • PARI
    {T(n,k) = my(A=[1,y],t); for(i=1,n, A=concat(A,0); t = ceil(sqrt(2*(#A)+9));
A[#A] = -polcoeff( sum(m=-t,t, (-1)^m*x^(m*(m+1)/2)*Ser(A)^m ), #A-1));polcoeff(A[n+1],k,y)}
for(n=0,12, for(k=0,n, print1( T(n,k),", "));print(""))

Formula

G.f. A(x) = Sum_{n>=0} x^n * Sum_{k=0..n} T(n,k)*y^k satisfies:
(1) x*y = Sum_{n=-oo..+oo} (-1)^n * x^(n*(n+1)/2) * A(x,y)^n.
(2) x*y*P(x) = Product_{n>=1} (1 - x^n*A(x,y)) * (1 - x^(n-1)/A(x,y)), where P(x) = Product_{n>=1} 1/(1 - x^n) is the partition function (A000041), due to the Jacobi triple product identity.

A354659 A diagonal of triangle A354650: a(n) = A354650(n,n+1), for n >= 0.

Original entry on oeis.org

1, 3, 30, 390, 5928, 98910, 1757688, 32683680, 628884300, 12428334215, 250940544738, 5156722096422, 107538413657010, 2270751678647100, 48464836803383400, 1044050265679857144, 22675350105240015204, 496034970650911331550, 10920742396832034391590
Offset: 0

Views

Author

Paul D. Hanna, Jun 02 2022

Keywords

Links

Crossrefs

Cf. A354649, A354650, A354658, A354660.

Programs

  • PARI
    {A354650(n,k) = my(A=[1+y]); for(i=1,n, A = concat(A,0);
A[#A] = polcoeff(y + sum(m=0,sqrtint(2*#A+9), (-1)^m * x^(m*(m-1)/2) * (1 - x^(2*m+1)) * Ser(A)^(m*(m+1)/2) ),#A-1) );
polcoeff(A[n+1],k,y)}
for(n=0,20,print1(A354650(n,n+1),", "))

Formula

a(n) = A354649(n,n+1), for n >= 0.
a(n) = A354650(n,n+1), for n >= 0.
a(n) ~ c * d^n / n^2, where d = 24.5759928778699928131449756... and c = 0.35661791857107638456206... - Vaclav Kotesovec, Mar 19 2023

A354660 a(n) = A354650(n,2*n), for n >= 0.

Original entry on oeis.org

1, 3, 15, 83, 486, 2937, 18109, 113220, 715122, 4552229, 29156985, 187683795, 1213110600, 7868238588, 51184173036, 333809308696, 2181842704602, 14288748463485, 93737673347185, 615889045662345, 4052198020223430, 26694405836621985, 176052003674681925
Offset: 0

Views

Author

Paul D. Hanna, Jun 02 2022

Keywords

Links

Crossrefs

Cf. A354649, A354650, A354658, A354659.

Programs

  • PARI
    {A354650(n,k) = my(A=[1+y]); for(i=1,n, A = concat(A,0);
A[#A] = polcoeff(y + sum(m=0,sqrtint(2*#A+9), (-1)^m * x^(m*(m-1)/2) * (1 - x^(2*m+1)) * Ser(A)^(m*(m+1)/2) ),#A-1) );
polcoeff(A[n+1],k,y)}
for(n=0,20,print1(A354650(n,2*n),", "))

Formula

a(n) = (-1)^(n+1) * A354649(n,2*n), for n >= 0.
a(n) = A354650(n,2*n), for n >= 0.
a(n) ~ 13 * 3^(3*n - 5/2) / (sqrt(Pi*n) * 2^(2*n)). - Vaclav Kotesovec, Mar 19 2023
Showing 1-5 of 5 results.

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

Content is available under The OEIS End-User License Agreement.