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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A355870 G.f. A(x,y) = Sum_{n>=0} x^n/(1-y)^(2*n+1) * Sum_{k=0..3*n} T(n,k)*y^k satisfies: y = Sum_{n=-oo..+oo} (-1)^n * x^(n*(n+1)/2) * A(x,y)^n.

Original entry on oeis.org

1, 0, 3, -3, 1, 0, 9, -18, 21, -15, 6, -1, 0, 22, -56, 116, -182, 196, -140, 64, -17, 2, 0, 51, -144, 496, -1329, 2436, -3148, 2934, -1971, 934, -297, 57, -5, 0, 108, -270, 1680, -7005, 18846, -36302, 52462, -57914, 49060, -31724, 15412, -5455, 1330, -200, 14, 0, 221, -381, 5647, -32760, 116068, -298976, 591690, -920249, 1138052, -1125135, 889253, -558740, 275744, -104672, 29524, -5833, 721, -42
Offset: 0

Views

Author

Paul D. Hanna, Jul 19 2022

Keywords

Comments

Row sums equal A000108, the Catalan numbers:
Sum_{k=0..3*n} T(n,k) = A000108(n) for n >= 0.
T(n,3*n) = (-1)^(n-1) * A000108(n-1) for n >= 1 (Catalan numbers).
Conjecture: T(n,1) = A000716(n) for n >= 1 (number of partitions of n into parts of 3 kinds).
The generating functions of some related sequences are given as follows.
(1) A(x,x) = Sum_{n>=0} A355351(n)*x^n.
(2) A(x,2*x) = Sum_{n>=0} A355352(n)*x^n.
(3) A(x,3*x) = Sum_{n>=0} A355353(n)*x^n.
(4) A(x,4*x) = Sum_{n>=0} A355354(n)*x^n.
(5) A(x,5*x) = Sum_{n>=0} A355355(n)*x^n.
(6) A(x,x^2) = Sum_{n>=0} A355356(n)*x^n.
(7) A(x^2,x) = Sum_{n>=0} A355357(n)*x^n.
(8) A(x,x*y) = Sum_{n>=0} x^n * Sum_{k=0..n} A355350(n,k) * y^k.
(9) 1/A(4*x,-1) = 2*Sum_{n>=0} A268300(n)*x^n.
(10) A(x,2) = -Sum_{n>=0} A355871(n)*x^n.
SPECIFIC VALUES.
(V.1) A(x,y) = -exp(-Pi) at x = exp(-2*Pi) and y = exp(Pi) * Pi^(1/4)/gamma(3/4).
(V.2) A(x,y) = -exp(-2*Pi) at x = exp(-4*Pi) and y = exp(2*Pi) * Pi^(1/4)/gamma(3/4) * (6 + 4*sqrt(2))^(1/4)/2.
(V.3) A(x,y) = -exp(-3*Pi) at x = exp(-6*Pi) and y = exp(3*Pi) * Pi^(1/4)/gamma(3/4) * (27 + 18*sqrt(3))^(1/4)/3.
(V.4) A(x,y) = -exp(-4*Pi) at x = exp(-8*Pi) and y = exp(4*Pi) * Pi^(1/4)/gamma(3/4) * (8^(1/4) + 2)/4.
(V.5) A(x,y) = -exp(-sqrt(3)*Pi) at x = exp(-2*sqrt(3)*Pi) and y = exp(sqrt(3)*Pi) * gamma(4/3)^(3/2)*3^(13/8)/(Pi*2^(2/3)).

Examples

			G.f.: A(x,y) = 1/(1-y) + x*(y^3 - 3*y^2 + 3*y)/(1-y)^3 + x^2*(-y^6 + 6*y^5 - 15*y^4 + 21*y^3 - 18*y^2 + 9*y)/(1-y)^5 + x^3*(2*y^9 - 17*y^8 + 64*y^7 - 140*y^6 + 196*y^5 - 182*y^4 + 116*y^3 - 56*y^2 + 22*y)/(1-y)^7 + x^4*(-5*y^12 + 57*y^11 - 297*y^10 + 934*y^9 - 1971*y^8 + 2934*y^7 - 3148*y^6 + 2436*y^5 - 1329*y^4 + 496*y^3 - 144*y^2 + 51*y)/(1-y)^9 + x^5*(14*y^15 - 200*y^14 + 1330*y^13 - 5455*y^12 + 15412*y^11 - 31724*y^10 + 49060*y^9 - 57914*y^8 + 52462*y^7 - 36302*y^6 + 18846*y^5 - 7005*y^4 + 1680*y^3 - 270*y^2 + 108*y)/(1-y)^11 + ...
where
y = ... + 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,
y = (1 - x*A(x,y))*(1 - 1/A(x,y))*(1-x) * (1 - x^2*A(x,y))*(1 - x/A(x,y))*(1-x^2) * (1 - x^3*A(x,y))*(1 - x^2/A(x,y))*(1-x^3) * (1 - x^4*A(x,y))*(1 - x^3/A(x,y))*(1-x^4) * ... * (1 - x^n*A(x,y))*(1 - x^(n-1)/A(x,y))*(1-x^n) * ...
This triangle of coefficients T(n,k) of x^n*y^k/(1-y)^(2*n+1) in A(x,y), for k = 0..3*n in row n, begins
n = 0: [1];
n = 1: [0, 3, -3, 1];
n = 2: [0, 9, -18, 21, -15, 6, -1];
n = 3: [0, 22, -56, 116, -182, 196, -140, 64, -17, 2];
n = 4: [0, 51, -144, 496, -1329, 2436, -3148, 2934, -1971, 934, -297, 57, -5];
n = 5: [0, 108, -270, 1680, -7005, 18846, -36302, 52462, -57914, 49060, -31724, 15412, -5455, 1330, -200, 14];
n = 6: [0, 221, -381, 5647, -32760, 116068, -298976, 591690, -920249, 1138052, -1125135, 889253, -558740, 275744, -104672, 29524, -5833, 721, -42];
n = 7: [0, 429, -63, 18281, -134985, 594399, -1941037, 4947447, -10062669, 16571700, -22316250, 24716922, -22564425, 16956135, -10435305, 5210319, -2078910, 647565, -151825, 25215, -2646, 132]; ...
The rightmost border equals the signed Catalan numbers (A000108) shifted right one place.
Column 1 appears to equal A000716 (ignoring the initial term).
Example: at y = x, we have the g.f. of A355351:
A(x,x) = 1/(1-x) + x*(3*x - 3*x^2 + x^3)/(1-x)^3 + x^2*(9*x - 18*x^2 + 21*x^3 - 15*x^4 + 6*x^5 - x^6)/(1-x)^5 + x^3*(22*x - 56*x^2 + 116*x^3 - 182*x^4 + 196*x^5 - 140*x^6 + 64*x^7 - 17*x^8 + 2*x^9)/(1-x)^7 + ... = 1 + x + 4*x^2 + 16*x^3 + 60*x^4 + 231*x^5 + 920*x^6 + 3819*x^7 + ... + A355351(n)*x^n + ...
where x = Sum_{n=-oo..+oo} (-1)^n * x^(n*(n+1)/2) * A(x,x)^n.

Links

Crossrefs

Cf. A000108 (row sums), A355871 (y=2).
Cf. A355350 (related triangle), A355351 (y=x), A355352 (y=2*x), A355353 (y=3*x), A355354 (y=4*x), A355355 (y=5*x), A355356 (y=x^2), A355357 (x=x^2,y=x).
Cf. A355360 (related triangle), A000716.

Programs

  • PARI
    {T(n,k) = my(A=[1/(1-y)],t); for(i=1,n, A=concat(A,0); t = ceil(sqrt(2*(#A)+9));
A[#A] = polcoeff( (y - sum(m=-t,t, (-1)^m * x^(m*(m+1)/2) * Ser(A)^m )), #A-1,x)/(1-y)^2);polcoeff(A[n+1]*(1-y)^(2*n+1),k,y)}
for(n=0,12, for(k=0,3*n, print1( T(n,k),", "));print(""))

Formula

G.f. A(x,y) = Sum_{n>=0} x^n/(1-y)^(2*n+1) * Sum_{k=0..3*n} T(n,k)*y^k satisfies:
(1) y = Sum_{n=-oo..+oo} (-1)^n * x^(n*(n+1)/2) * A(x,y)^n.
(2) y = Product_{n>=1} (1 - x^n*A(x,y)) * (1 - x^(n-1)/A(x,y)) * (1 - x^n), by the Jacobi triple product identity.

A355349 G.f. A(x) satisfies: 2 = Sum_{n=-oo..+oo} (-x)^(n*(n-1)/2) * A(x)^(n^2).

Original entry on oeis.org

1, 2, 10, 76, 678, 6608, 68170, 731638, 8084692, 91361298, 1050937008, 12264790410, 144856757032, 1728197200206, 20796217437806, 252117655811806, 3076371017010508, 37753163861001044, 465657991700212170, 5769586313420410060, 71777257553636752194
Offset: 0

Views

Author

Paul D. Hanna, Aug 02 2022

Keywords

Examples

			G.f.: A(x) = 1 + 2*x + 10*x^2 + 76*x^3 + 678*x^4 + 6608*x^5 + 68170*x^6 + 731638*x^7 + 8084692*x^8 + 91361298*x^9 + 1050937008*x^10 + ...
where
2 = ... + x^6*A(x)^9 - x^3*A(x)^4 - x*A(x) + A(x) + 1 - x*A(x) - x^3*A(x)^4 + x^6*A(x)^9 + x^10*A(x)^16 - x^15*A(x)^25 - x^21*A(x)^36 ++-- ... (-x)^(n*(n-1)/2) * A(x)^(n^2) + ...

Links

Crossrefs

Cf. A355871, A354248, A354662.

Programs

  • Mathematica
    (* Calculation of constant d: *) 1/r /. FindRoot[{r*s^2*QPochhammer[-1/s, -r*s^2]* QPochhammer[1/(r*s), -r*s^2]* QPochhammer[-r*s^2]/((1 + s)*(-1 + r*s)) == 2, (4 + 2*s - 2*r*s)/(r*s^2) + 2*r*s^2*QPochhammer[1/(r*s), -r*s^2]* QPochhammer[-r*s^2] * Derivative[0, 1][QPochhammer][-1/s, -r*s^2] + 4*(1 + s)*(-1 + r*s)* Derivative[0, 1][QPochhammer][1/(r*s), -r*s^2] / QPochhammer[1/(r*s), -r*s^2] + 2*(1 + s)*(-1 + r*s) * ((2*QPolyGamma[0, 1, -r*s^2] - QPolyGamma[0, Log[-1/s]/Log[-r*s^2], -r*s^2] - QPolyGamma[0, Log[1/(r*s)]/Log[-r*s^2], -r*s^2])/(r*s^2* Log[-r*s^2]) + (2*Derivative[0, 1][QPochhammer][-r*s^2, -r*s^2])/ QPochhammer[-r*s^2]) == 0}, {r, 1/12}, {s, 3/2}, WorkingPrecision -> 70] (* Vaclav Kotesovec, Jan 31 2024 *)
  • PARI
    {a(n) = my(A=[1]); for(i=1,n, A=concat(A,0); M=ceil(sqrt(2*n));
A[#A] = -polcoeff( sum(m=-M,M, (-x)^(m*(m-1)/2) * Ser(A)^(m^2)),#A-1));A[n+1]}
for(n=0,20,print1(a(n),", "))

Formula

G.f. A(x) satisfies:
(1) 2 = Sum_{n=-oo..+oo} (-x)^(n*(n-1)/2) * A(x)^(n^2).
(2) 2 = Product_{n>=1} (1 - (-x)^n*A(x)^(2*n)) * (1 + (-x)^(n-1)*A(x)^(2*n-1)) * (1 + (-x)^n*A(x)^(2*n-1)), by the Jacobi triple product identity.
From Vaclav Kotesovec, Jan 31 2024: (Start)
Formula (2) can be rewritten as the functional equation QPochhammer(-x*y^2) * QPochhammer(1/(x*y), -x*y^2)/(1 - 1/(x*y)) * QPochhammer(-1/y, -x*y^2)/(1 + 1/y) = 2.
a(n) ~ c * d^n / n^(3/2), where d = 13.4235502463317299100709807099986120871056759637108569... and c = 0.1810118197770993236368884418746617625188562698029... (End)
Showing 1-2 of 2 results.

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