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 1-8 of 8 results.

A355357 G.f. A(x) satisfies: x = Sum_{n=-oo..+oo} (-1)^n * x^(n*(n+1)) * A(x)^n.

Original entry on oeis.org

1, 1, 1, 4, 7, 20, 43, 110, 262, 674, 1684, 4397, 11320, 29938, 78641, 210044, 559724, 1507563, 4060585, 11016027, 29919220, 81673846, 223307300, 612851316, 1684816018, 4645243490, 12829177587, 35513736868, 98465916370, 273531234027, 760966444416
Offset: 0

Views

Author

Paul D. Hanna, Jun 29 2022

Keywords

Comments

a(n) = Sum_{k=0..floor(n/2)} A355350(n-k,n-2*k) for n >= 0.
a(n) = A359720(n,0), for n >= 0.

Examples

			G.f.: A(x) = 1 + x + x^2 + 4*x^3 + 7*x^4 + 20*x^5 + 43*x^6 + 110*x^7 + 262*x^8 + 674*x^9 + 1684*x^10 + 4397*x^11 + 11320*x^12 + ...
where
x = ... + x^12/A(x)^4 - x^6/A(x)^3 + x^2/A(x)^2 - 1/A(x) + 1 - x^2*A(x) + x^6*A(x)^2 - x^12*A(x)^3 + x^20*A(x)^4 -+ ...
also,
x*P(x^2) = (1 - x^2*A(x))*(1 - 1/A(x)) * (1 - x^4*A(x))*(1 - x^2/A(x)) * (1 - x^6*A(x))*(1 - x^4/A(x)) * (1 - x^8*A(x))*(1 - x^6/A(x)) * ...
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 + 56*x^11 + 77*x^12 + ... + A000041(n)*x^n + ...

Links

Crossrefs

Cf. A355350, A355351, A355352, A355353, A355354, A355355, A355356.
Cf. A359720, A357221, A357222, A357223, A357224, A357225, A357226.

Programs

  • Mathematica
    (* Calculation of constants {d,c}: *) {1/r, Sqrt[-Log[r] * ((-1 + r) * QPochhammer[1/r, r^2] * (-2*Log[r] + (-1 + r)*(Log[1 - r^2] - Log[r - r^3]) + (-1 + r) * QPolyGamma[0, -1/2, r^2] - (-1 + r)*QPolyGamma[0, 1, r^2]) + 4*(-1 + r)^2 * r^2 * Log[r] * Derivative[0, 1][QPochhammer][1/r, r^2] + 2*r^3 * Log[r] * QPochhammer[1/r, r^2]^3 * Derivative[0, 1][QPochhammer][r^2, r^2]) / (Pi*r^2* QPochhammer[1/r, r^2] * (-4*r*Log[r]^2 + (-1 + r)^2 * QPolyGamma[1, -1/2, r^2]))]} /. FindRoot[ 1/QPochhammer[r^2] == (r*QPochhammer[1/r, r^2]^2)/(-1 + r)^2, {r, 1/3}, WorkingPrecision -> 70] (* Vaclav Kotesovec, Feb 01 2024 *)
  • PARI
    {a(n) = my(A=[1,1],t); for(i=1,n, A=concat(A,0); t = ceil(sqrt(n+4));
A[#A] = -polcoeff( sum(m=-t,t, (-1)^m*x^(m*(m+1))*Ser(A)^m ), #A-1));A[n+1]}
for(n=0,30,print1(a(n),", "))

Formula

G.f. A(x) satisfies:
(1) x = Sum_{n=-oo..+oo} (-1)^n * x^(n*(n+1)) * A(x)^n.
(2) x*P(x^2) = Product_{n>=1} (1 - x^(2*n)*A(x)) * (1 - x^(2*n-2)/A(x)), where P(x) = Product_{n>=1} 1/(1 - x^n) is the partition function (A000041), due to the Jacobi triple product identity.
From Vaclav Kotesovec, Feb 01 2024: (Start)
Formula (2) can be rewritten as the functional equation x/QPochhammer(x^2) = QPochhammer(y, x^2)/(1 - y) * QPochhammer(1/(x^2*y), x^2)/(1 - 1/(x^2*y)).
a(n) ~ c * d^n / n^(3/2), where d = 2.92005174190265697439941308343193651904071627244119127019370275824199... and c = 1.4709989760845501303394202030872391136773745007487301056274536584990... (End)

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.

A355351 G.f. A(x) satisfies: x = Sum_{n=-oo..+oo} (-1)^n * x^(n*(n+1)/2) * A(x)^n.

Original entry on oeis.org

1, 1, 4, 16, 60, 231, 920, 3819, 16365, 71792, 320219, 1446517, 6602975, 30415725, 141231704, 660431602, 3107519738, 14701758926, 69891556656, 333700223891, 1599475107712, 7693580712200, 37125486197570, 179675330190428, 871910824853956, 4241603521253775
Offset: 0

Views

Author

Paul D. Hanna, Jun 29 2022

Keywords

Comments

a(n) = Sum_{k=0..n} A355350(n,k) for n >= 0.

Examples

			G.f.: A(x) = 1 + x + 4*x^2 + 16*x^3 + 60*x^4 + 231*x^5 + 920*x^6 + 3819*x^7 + 16365*x^8 + 71792*x^9 + 320219*x^10 + 1446517*x^11 + ...
where
x = ... - x^10/A(x)^5 + x^6/A(x)^4 - x^3/A(x)^3 + x/A(x)^2 - 1/A(x) + 1 - x*A(x) + x^3*A(x)^2 - x^6*A(x)^3 + x^10*A(x)^4 -+ ...
also,
x*P(x) = (1 - x*A(x))*(1 - 1/A(x)) * (1 - x^2*A(x))*(1 - x/A(x)) * (1 - x^3*A(x))*(1 - x^2/A(x)) * (1 - x^4*A(x))*(1 - x^3/A(x)) * ...
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 + 56*x^11 + 77*x^12 + ... + A000041(n)*x^n + ...

Links

Crossrefs

Cf. A355350, A355352, A355353, A355354, A355355, A355356, A355357.

Programs

  • Mathematica
    (* Calculation of constants {d,c}: *) {1/r, s*Sqrt[(-1 + s)*(-1 + r*s) * Log[r]* ((-1 + s)*(-1 + r*s) * QPolyGamma[0, 1, r] - r*(-1 + s)*(-1 + r*s)*Log[r]* Derivative[0, 1][QPochhammer][r, r] / QPochhammer[r] + r*s*Log[r] * QPochhammer[r] * QPochhammer[s, r] * Derivative[0, 1][QPochhammer][1/(r*s), r] + (-1 + r*s) * ((1 - s) * QPolyGamma[0, Log[s]/Log[r], r] + Log[r] * (-1 - (r*(-1 + s) * Derivative[0, 1][QPochhammer][s, r]) / QPochhammer[s, r]))) / (-s*(1 + r - 4*r*s + r*(1 + r)*s^2) * Log[r]^2 + (-1 + s)^2 * (-1 + r*s)^2 * QPolyGamma[1, Log[s]/Log[r], r] + (-1 + s)^2 * (-1 + r*s)^2 * QPolyGamma[1, -Log[r*s]/Log[r], r]) / (2*Pi)]} /. FindRoot[{1/QPochhammer[r] + s*QPochhammer[1/(r*s), r] * QPochhammer[s, r] / ((-1 + s) * (-1 + r*s)) == 0, (-1 + r*s^2)*Log[r] + (-1 + s) * (-1 + r*s) * (QPolyGamma[0, Log[s]/Log[r], r] - QPolyGamma[0, -Log[r*s] / Log[r], r]) == 0}, {r, 1/5}, {s, 2}, WorkingPrecision -> 70] (* Vaclav Kotesovec, Feb 01 2024 *)
  • PARI
    {a(n) = my(A=[1,1],t); for(i=1,n, A=concat(A,0); t = ceil(sqrt(2*n+9));
A[#A] = -polcoeff( sum(m=-t,t, (-1)^m*x^(m*(m+1)/2)*Ser(A)^m ), #A-1));A[n+1]}
for(n=0,30,print1(a(n),", "))

Formula

G.f. A(x) satisfies:
(1) x = Sum_{n=-oo..+oo} (-1)^n * x^(n*(n+1)/2) * A(x)^n.
(2) x*P(x) = Product_{n>=1} (1 - x^n*A(x)) * (1 - x^(n-1)/A(x)), where P(x) = Product_{n>=1} 1/(1 - x^n) is the partition function (A000041), due to the Jacobi triple product identity.
From Vaclav Kotesovec, Feb 01 2024: (Start)
Formula (2) can be rewritten as the functional equation x/QPochhammer(x) = QPochhammer(y, x)/(1 - y) * QPochhammer(1/(x*y), x)/(1 - 1/(x*y)).
a(n) ~ c * d^n / n^(3/2), where d = 5.163920888936085556632234304058129... and c = 0.824708825453794494929019119272... (End)

A355353 G.f. A(x) satisfies: 3*x = Sum_{n=-oo..+oo} (-1)^n * x^(n*(n+1)/2) * A(x)^n.

Original entry on oeis.org

1, 3, 18, 108, 660, 4275, 29106, 205377, 1485279, 10943424, 81866493, 620316297, 4751289063, 36727782675, 286153810542, 2244799306134, 17715992048886, 140560480602810, 1120518766292436, 8970573523101477, 72091628161825608, 581375787259765554, 4703286596619094686
Offset: 0

Views

Author

Paul D. Hanna, Jun 29 2022

Keywords

Comments

a(n) = Sum_{k=0..n} A355350(n,k) * 3^k for n >= 0.

Examples

			G.f.: A(x) = 1 + 3*x + 18*x^2 + 108*x^3 + 660*x^4 + 4275*x^5 + 29106*x^6 + 205377*x^7 + 1485279*x^8 + 10943424*x^9 + 81866493*x^10 + ...
where
3*x = ... - x^10/A(x)^5 + x^6/A(x)^4 - x^3/A(x)^3 + x/A(x)^2 - 1/A(x) + 1 - x*A(x) + x^3*A(x)^2 - x^6*A(x)^3 + x^10*A(x)^4 -+ ...
also,
3*x*P(x) = (1 - x*A(x))*(1 - 1/A(x)) * (1 - x^2*A(x))*(1 - x/A(x)) * (1 - x^3*A(x))*(1 - x^2/A(x)) * (1 - x^4*A(x))*(1 - x^3/A(x)) * ...
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 + 56*x^11 + 77*x^12 + ... + A000041(n)*x^n + ...

Links

Crossrefs

Cf. A355350, A355351, A355352, A355354, A355355, A355356, A355357.

Programs

  • PARI
    {a(n) = my(A=[1,3],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));A[n+1]}
for(n=0,30,print1(a(n),", "))

Formula

G.f. A(x) satisfies:
(1) 3*x = Sum_{n=-oo..+oo} (-1)^n * x^(n*(n+1)/2) * A(x)^n.
(2) 3*x*P(x) = Product_{n>=1} (1 - x^n*A(x)) * (1 - x^(n-1)/A(x)), where P(x) = Product_{n>=1} 1/(1 - x^n) is the partition function (A000041), due to the Jacobi triple product identity.

A355354 G.f. A(x) satisfies: 4*x = Sum_{n=-oo..+oo} (-1)^n * x^(n*(n+1)/2) * A(x)^n.

Original entry on oeis.org

1, 4, 28, 196, 1416, 10860, 87392, 727188, 6196212, 53783336, 474011756, 4231158016, 38174676188, 347566170384, 3189295781780, 29465038957708, 273851282010308, 2558703740102840, 24019990008557160, 226444571054525156, 2142925363606256584, 20349477565111498148
Offset: 0

Views

Author

Paul D. Hanna, Jun 29 2022

Keywords

Comments

a(n) = Sum_{k=0..n} A355350(n,k) * 4^k for n >= 0.

Examples

			G.f.: A(x) = 1 + 4*x + 28*x^2 + 196*x^3 + 1416*x^4 + 10860*x^5 + 87392*x^6 + 727188*x^7 + 6196212*x^8 + 53783336*x^9 + 474011756*x^10 + ...
where
4*x = ... - x^10/A(x)^5 + x^6/A(x)^4 - x^3/A(x)^3 + x/A(x)^2 - 1/A(x) + 1 - x*A(x) + x^3*A(x)^2 - x^6*A(x)^3 + x^10*A(x)^4 -+ ...
also,
4*x*P(x) = (1 - x*A(x))*(1 - 1/A(x)) * (1 - x^2*A(x))*(1 - x/A(x)) * (1 - x^3*A(x))*(1 - x^2/A(x)) * (1 - x^4*A(x))*(1 - x^3/A(x)) * ...
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 + 56*x^11 + 77*x^12 + ... + A000041(n)*x^n + ...

Links

Crossrefs

Cf. A355350, A355351, A355352, A355353, A355355, A355356, A355357.

Programs

  • PARI
    {a(n) = my(A=[1,4],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));A[n+1]}
for(n=0,30,print1(a(n),", "))

Formula

G.f. A(x) satisfies:
(1) 4*x = Sum_{n=-oo..+oo} (-1)^n * x^(n*(n+1)/2) * A(x)^n.
(2) 4*x*P(x) = Product_{n>=1} (1 - x^n*A(x)) * (1 - x^(n-1)/A(x)), where P(x) = Product_{n>=1} 1/(1 - x^n) is the partition function (A000041), due to the Jacobi triple product identity.

A355355 G.f. A(x) satisfies: 5*x = Sum_{n=-oo..+oo} (-1)^n * x^(n*(n+1)/2) * A(x)^n.

Original entry on oeis.org

1, 5, 40, 320, 2660, 23455, 216540, 2064055, 20137945, 200134600, 2019406895, 20635313325, 213109960895, 2220820915065, 23323755734820, 246616999661690, 2623193780773530, 28049464032800110, 301340494687086960, 3251017466141039095, 35207152686408604400
Offset: 0

Views

Author

Paul D. Hanna, Jun 29 2022

Keywords

Comments

a(n) = Sum_{k=0..n} A355350(n,k) * 5^k for n >= 0.

Examples

			G.f.: A(x) = 1 + 5*x + 40*x^2 + 320*x^3 + 2660*x^4 + 23455*x^5 + 216540*x^6 + 2064055*x^7 + 20137945*x^8 + 200134600*x^9 + 2019406895*x^10 + ...
where
5*x = ... - x^10/A(x)^5 + x^6/A(x)^4 - x^3/A(x)^3 + x/A(x)^2 - 1/A(x) + 1 - x*A(x) + x^3*A(x)^2 - x^6*A(x)^3 + x^10*A(x)^4 -+ ...
also,
5*x*P(x) = (1 - x*A(x))*(1 - 1/A(x)) * (1 - x^2*A(x))*(1 - x/A(x)) * (1 - x^3*A(x))*(1 - x^2/A(x)) * (1 - x^4*A(x))*(1 - x^3/A(x)) * ...
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 + 56*x^11 + 77*x^12 + ... + A000041(n)*x^n + ...

Links

Crossrefs

Cf. A355350, A355351, A355352, A355353, A355354, A355356, A355357.

Programs

  • PARI
    {a(n) = my(A=[1,5],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));A[n+1]}
for(n=0,30,print1(a(n),", "))

Formula

G.f. A(x) satisfies:
(1) 5*x = Sum_{n=-oo..+oo} (-1)^n * x^(n*(n+1)/2) * A(x)^n.
(2) 5*x*P(x) = Product_{n>=1} (1 - x^n*A(x)) * (1 - x^(n-1)/A(x)), where P(x) = Product_{n>=1} 1/(1 - x^n) is the partition function (A000041), due to the Jacobi triple product identity.

A355356 G.f. A(x) satisfies: x^2 = Sum_{n=-oo..+oo} (-1)^n * x^(n*(n+1)/2) * A(x)^n.

Original entry on oeis.org

1, 0, 1, 3, 10, 28, 79, 216, 603, 1702, 4933, 14620, 44287, 136352, 424858, 1334162, 4211572, 13344072, 42412667, 135217722, 432483522, 1387929369, 4469341807, 14439523193, 46795072968, 152076428228, 495460089510, 1617787324674, 5292984017236, 17348743335252
Offset: 0

Views

Author

Paul D. Hanna, Jun 29 2022

Keywords

Comments

a(n) = Sum_{k=0..floor(n/2)} A355350(n-k,k) for n >= 0.

Examples

			G.f.: A(x) = 1 + x^2 + 3*x^3 + 10*x^4 + 28*x^5 + 79*x^6 + 216*x^7 + 603*x^8 + 1702*x^9 + 4933*x^10 + 14620*x^11 + 44287*x^12 + ...
where
x^2 = ... - x^10/A(x)^5 + x^6/A(x)^4 - x^3/A(x)^3 + x/A(x)^2 - 1/A(x) + 1 - x*A(x) + x^3*A(x)^2 - x^6*A(x)^3 + x^10*A(x)^4 -+ ...
also,
x^2*P(x) = (1 - x*A(x))*(1 - 1/A(x)) * (1 - x^2*A(x))*(1 - x/A(x)) * (1 - x^3*A(x))*(1 - x^2/A(x)) * (1 - x^4*A(x))*(1 - x^3/A(x)) * ...
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 + 56*x^11 + 77*x^12 + ... + A000041(n)*x^n + ...

Crossrefs

Cf. A355350, A355351, A355352, A355353, A355354, A355355, A355357.

Programs

  • PARI
    {a(n) = my(A=[1,0,1],t); for(i=1,n, A=concat(A,0); t = ceil(sqrt(2*n+9));
A[#A] = polcoeff( x^2 - sum(m=-t,t, (-1)^m*x^(m*(m+1)/2)*Ser(A)^m ), #A-1));A[n+1]}
for(n=0,30,print1(a(n),", "))

Formula

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

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.
Showing 1-8 of 8 results.

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