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A355357

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

+10
21

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

(list; graph; refs; listen; history; text; internal format)

OFFSET

0,4

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.

LINKS

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

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)

EXAMPLE

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 + ...

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 *)

PROG

(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), ", "))

CROSSREFS

Cf. A355350, A355351, A355352, A355353, A355354, A355355, A355356.

Cf. A359720, A357221, A357222, A357223, A357224, A357225, A357226.

KEYWORD

nonn

AUTHOR

Paul D. Hanna, Jun 29 2022

STATUS

approved

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.

+10
11

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

(list; table; graph; refs; listen; history; text; internal format)

OFFSET

0,5

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.

LINKS

Paul D. Hanna, Table of n, a(n) for n = 0..5150

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.

EXAMPLE

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 + ...

PROG

(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(""))

CROSSREFS

Cf. A355351 (row sums), A355352, A355353, A355354, A355355.

Cf. A355356, A355357, A354658 (central terms).

Cf. A354645, A354650 (related table), A000041, A000716, A023005.

KEYWORD

nonn,tabl

AUTHOR

Paul D. Hanna, Jun 29 2022

STATUS

approved

A355351

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

+10
9

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

(list; graph; refs; listen; history; text; internal format)

OFFSET

0,3

COMMENTS

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

LINKS

Paul D. Hanna, Table of n, a(n) for n = 0..600

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)

EXAMPLE

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 + ...

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 *)

PROG

(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), ", "))

CROSSREFS

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

KEYWORD

nonn

AUTHOR

Paul D. Hanna, Jun 29 2022

STATUS

approved

A355352

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

+10
9

1, 2, 10, 50, 248, 1294, 7092, 40426, 236698, 1412860, 8561906, 52546920, 326011118, 2041512624, 12886608654, 81908498582, 523780469070, 3367399778356, 21752611767804, 141118852010146, 919035717462824, 6006146649948722, 39376700396145616, 258907024677687808

(list; graph; refs; listen; history; text; internal format)

OFFSET

0,2

COMMENTS

a(n) = Sum_{k=0..n} A355350(n,k) * 2^k for n >= 0.

LINKS

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

FORMULA

G.f. A(x) satisfies:

(1) 2*x = Sum_{n=-oo..+oo} (-1)^n * x^(n*(n+1)/2) * A(x)^n.

(2) 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.

EXAMPLE

G.f.: A(x) = 1 + 2*x + 10*x^2 + 50*x^3 + 248*x^4 + 1294*x^5 + 7092*x^6 + 40426*x^7 + 236698*x^8 + 1412860*x^9 + 8561906*x^10 + ...

where

2*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,

2*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 + ...

PROG

(PARI) {a(n) = my(A=[1, 2], 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), ", "))

CROSSREFS

Cf. A355350, A355351, A355353, A355354, A355355, A355356, A355357.

KEYWORD

nonn

AUTHOR

Paul D. Hanna, Jun 29 2022

STATUS

approved

A355354

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

+10
9

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

(list; graph; refs; listen; history; text; internal format)

OFFSET

0,2

COMMENTS

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

LINKS

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

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.

EXAMPLE

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 + ...

PROG

(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), ", "))

CROSSREFS

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

KEYWORD

nonn

AUTHOR

Paul D. Hanna, Jun 29 2022

STATUS

approved

A355355

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

+10
9

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

(list; graph; refs; listen; history; text; internal format)

OFFSET

0,2

COMMENTS

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

LINKS

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

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.

EXAMPLE

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 + ...

PROG

(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), ", "))

CROSSREFS

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

KEYWORD

nonn

AUTHOR

Paul D. Hanna, Jun 29 2022

STATUS

approved

A355356

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

+10
8

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

(list; graph; refs; listen; history; text; internal format)

OFFSET

0,4

COMMENTS

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

LINKS

Table of n, a(n) for n=0..29.

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.

EXAMPLE

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 + ...

PROG

(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), ", "))

CROSSREFS

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

KEYWORD

nonn

AUTHOR

Paul D. Hanna, Jun 29 2022

STATUS

approved

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.

+10
3

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

(list; graph; refs; listen; history; text; internal format)

OFFSET

0,3

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)).

LINKS

Paul D. Hanna, Table of n, a(n) for n = 0..3875

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.

EXAMPLE

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.

PROG

(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(""))

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.

KEYWORD

sign,tabf

AUTHOR

Paul D. Hanna, Jul 19 2022

STATUS

approved

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