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.

A356504 a(n) = A356500(2*n, 2*n+1) for n >= 0.

Original entry on oeis.org

1, 4, 84, 2568, 94390, 3863496, 169713208, 7836945872, 375608185758, 18527792412380, 935129979113044, 48088668037229040, 2511680568602631894, 132918633258508425944, 7113508747197660153120, 384416086900675623039520, 20951080869890118976964642
Offset: 0

Views

Author

Paul D. Hanna, Aug 09 2022

Keywords

Comments

Triangle A356500 describes the coefficients in G(x,y) that satisfies: y = Sum_{n=-oo..+oo} (-x)^(n^2) * G(x,y)^((n-1)^2).

Crossrefs

Cf. A356500, A356505, A356506.

Programs

  • PARI
    {A356500(n,k) = my(A=[y],M); for(i=1,n, A = concat(A,0); M = ceil(sqrt(n+1));
A[#A] = -polcoeff( sum(m=-M,M, (-x)^(m^2)*Ser(A)^((m-1)^2)), #A-1)); polcoeff(A[n+1],k,y) }
for(n=0,20, print1( A356500(2*n, 2*n+1),", "))

A356505 a(n) = A356500(2*n+1, 2*n) for n >= 0.

Original entry on oeis.org

1, 6, 135, 4284, 160965, 6684744, 296679006, 13805453160, 665611197093, 32988925715610, 1671463040525586, 86231285273788180, 4516133521439246825, 239551205985729110664, 12846081444122599438850, 695428535332816056597520, 37960416340437631597631877
Offset: 0

Views

Author

Paul D. Hanna, Aug 09 2022

Keywords

Comments

Triangle A356500 describes the coefficients in G(x,y) that satisfies: y = Sum_{n=-oo..+oo} (-x)^(n^2) * G(x,y)^((n-1)^2).

Crossrefs

Cf. A356500, A356504, A356506.

Programs

  • PARI
    {A356500(n,k) = my(A=[y],M); for(i=1,n, A = concat(A,0); M = ceil(sqrt(n+1));
A[#A] = -polcoeff( sum(m=-M,M, (-x)^(m^2)*Ser(A)^((m-1)^2)), #A-1)); polcoeff(A[n+1],k,y) }
for(n=0,20, print1( A356500(2*n+1, 2*n),", "))

A356506 a(n) = A356500(3*n, n+1) for n >= 0.

Original entry on oeis.org

1, 6, 120, 3250, 103020, 3587696, 133101836, 5167564380, 207615129579, 8567305854998, 361201849117032, 15498967122249676, 674906101555736960, 29757755664623031984, 1326196334421645347368, 59655785739373960058296, 2705420198806474232850741
Offset: 0

Views

Author

Paul D. Hanna, Aug 09 2022

Keywords

Comments

Triangle A356500 describes the coefficients in G(x,y) that satisfies: y = Sum_{n=-oo..+oo} (-x)^(n^2) * G(x,y)^((n-1)^2).

Crossrefs

Cf. A356500, A356504, A356505.

Programs

  • PARI
    {A356500(n,k) = my(A=[y],M); for(i=1,n, A = concat(A,0); M = ceil(sqrt(n+1));
A[#A] = -polcoeff( sum(m=-M,M, (-x)^(m^2)*Ser(A)^((m-1)^2)), #A-1)); polcoeff(A[n+1],k,y) }
for(n=0,20, print1( A356500(3*n, n+1),", "))

A361050 Expansion of g.f. A(x,y) satisfying y/x = Sum_{n=-oo..+oo} x^(n*(3*n+1)/2) * (A(x,y)^(3*n) - 1/A(x,y)^(3*n+1)), as a triangle read by rows.

Original entry on oeis.org

1, 0, 1, 0, 5, 4, 0, 18, 40, 22, 0, 55, 244, 335, 140, 0, 149, 1160, 2924, 2875, 969, 0, 371, 4688, 19090, 32745, 25081, 7084, 0, 867, 16848, 103110, 272250, 352814, 221397, 53820, 0, 1923, 55332, 485356, 1839075, 3565548, 3709244, 1971775, 420732, 0, 4086, 169048, 2054520, 10674985, 28909300, 44146487, 38344384, 17682895, 3362260
Offset: 1

Views

Author

Paul D. Hanna, Mar 18 2023

Keywords

Comments

A359921(n) = Sum_{k=0..n-1} T(n,k) for n >= 1.
A359924(n) = Sum_{k=0..n-1} T(n,k) * 2^k for n >= 1.
A361051(n) = Sum_{k=0..n-1} T(n,k) * 3^k for n >= 1.
A361052(n) = Sum_{k=0..n-1} T(n,k) * 4^k for n >= 1.
A361538(n) = T(2*n-1,n-1) for n >= 1.
A360191(n) = T(n+2,1) for n >= 0.
A361535(n) = T(n+3,2)/4 for n >= 0.
A002293(n) = T(n+1,n) for n >= 0.

Examples

			G.f.: A(x,y) = x + y*x^2 + (5*y + 4*y^2)*x^3 + (18*y + 40*y^2 + 22*y^3)*x^4 + (55*y + 244*y^2 + 335*y^3 + 140*y^4)*x^5 + (149*y + 1160*y^2 + 2924*y^3 + 2875*y^4 + 969*y^5)*x^6 + (371*y + 4688*y^2 + 19090*y^3 + 32745*y^4 + 25081*y^5 + 7084*y^6)*x^7 + (867*y + 16848*y^2 + 103110*y^3 + 272250*y^4 + 352814*y^5 + 221397*y^6 + 53820*y^7)*x^8 + (1923*y + 55332*y^2 + 485356*y^3 + 1839075*y^4 + 3565548*y^5 + 3709244*y^6 + 1971775*y^7 + 420732*y^8)*x^9 + (4086*y + 169048*y^2 + 2054520*y^3 + 10674985*y^4 + 28909300*y^5 + 44146487*y^6 + 38344384*y^7 + 17682895*y^8 + 3362260*y^9)*x^10 + ...
This triangle of coefficients T(n,k) of x^n*y^k, n >= 1, k = 0..n-1, in g.f. A(x,y) begins:
1;
0, 1;
0, 5, 4;
0, 18, 40, 22;
0, 55, 244, 335, 140;
0, 149, 1160, 2924, 2875, 969;
0, 371, 4688, 19090, 32745, 25081, 7084;
0, 867, 16848, 103110, 272250, 352814, 221397, 53820;
0, 1923, 55332, 485356, 1839075, 3565548, 3709244, 1971775, 420732;
0, 4086, 169048, 2054520, 10674985, 28909300, 44146487, 38344384, 17682895, 3362260;
0, 8374, 486500, 7984667, 55085875, 199363606, 417661860, 525322468, 391561335, 159463876, 27343888;
0, 16634, 1331056, 28909580, 258486830, 1211896230, 3335033317, 5680806120, 6069336891, 3961602925, 1444601027, 225568798;
...

Links

Crossrefs

Cf. A360191 (column 1), A361535 (column 2), A002293 (diagonal), A361538 (central terms).
Cf. A359921 (y=1), A359924 (y=2), A361051 (y=3), A361052 (y=4).
Cf. A002293, A356500 (related table), A361550 (related triangle).

Programs

  • PARI
    {T(n,k) = my(A=[0, 1]); for(i=1, n, A = concat(A, 0);
A[#A] = polcoeff(y/x - prod(m=1, #A, (1 - x^m) * (1 - x^m*Ser(A)) * (1 - x^(m-1)/Ser(A)) * (1 - x^(2*m-1)*Ser(A)^2) * (1 - x^(2*m-1)/Ser(A)^2) ), #A-4) );
polcoeff(polcoeff(H=Ser(A),n,x),k,y)}
for(n=1, 12, for(k=0,n-1, print1(T(n,k), ", "));print(""))

Formula

G.f. A(x,y) = Sum_{n>=1} Sum_{k=0..n-1} T(n,k)*x^n*y^k satisfies the following.
(1) y/x = Sum_{n=-oo..+oo} x^(n*(3*n+1)/2) * (A(x,y)^(3*n) - 1/A(x,y)^(3*n+1)).
(2) y/x = Product_{n>=1} (1 - x^n) * (1 - x^n*A(x,y)) * (1 - x^(n-1)/A(x,y)) * (1 - x^(2*n-1)*A(x,y)^2) * (1 - x^(2*n-1)/A(x,y)^2), by the Watson quintuple product identity.
(3) Sum_{n>=0} T(n+2,1) * x^n = 1 / Product_{n>=1} (1 - x^n)^3 * (1 - x^(2*n-1))^2, which is the g.f. of A360191.
(4) Sum_{n>=0} T(n+3,2) * x^n = 4*F(x) where F(x) = 1/Product_{n>=1} (1 - x^n)^6 * (1 - x^(2*n-1))^4, which is the g.f. of A361535.
(5) Sum_{n>=0} T(n+1,n) * x^n = D(x) where D(x) = 1 + x*D(x)^4 is the g.f. of A002293.
(6) T(n+1,n) = binomial(4*n, n)/(3*n + 1) for n >= 0.

A356502 G.f. A(x) satisfies: 2 = Sum_{n=-oo..+oo} (-x)^(n^2) * A(x)^((n-1)^2).

Original entry on oeis.org

2, 17, 544, 24344, 1261702, 71159152, 4240009152, 262584135640, 16734002688722, 1090225325371424, 72285357987696768, 4861658409827006872, 330874470176939132844, 22744684876060771599568, 1576898258893213475814464, 110136698483814852518084528, 7742091796859524187452564262
Offset: 0

Views

Author

Paul D. Hanna, Aug 09 2022

Keywords

Examples

			G.f.: A(x) = 2 + 17*x + 544*x^2 + 24344*x^3 + 1261702*x^4 + 71159152*x^5 + 4240009152*x^6 + 262584135640*x^7 + 16734002688722*x^8 + ...
such that A = A(x) satisfies
2 = ... + x^16*A^25 - x^9*A^16 + x^4*A^9 - x*A^4 + A - x + x^4*A - x^9*A^4 + x^16*A^9 - x^25*A^16 +- ... + (-x)^(n^2) * A(x,y)^((n-1)^2) + ...

Links

Crossrefs

Cf. A356500, A354248, A356503, A356504.

Programs

  • Mathematica
    (* Calculation of constant d: *) 1/r /. FindRoot[{k == r^4*s^2 * QPochhammer[1/(r^3*s), r^2*s^2] * QPochhammer[r/s, r^2*s^2] * QPochhammer[r^2*s^2, r^2*s^2]/((r - s)*(-1 + r^3*s)), 1/r^3*(k*(1 + r^4 - 2*r/s) + 2*r^6*s^3*QPochhammer[r/s, r^2*s^2] * QPochhammer[r^2*s^2, r^2*s^2]* Derivative[0, 1][QPochhammer][1/(r^3*s), r^2*s^2] + 2*k*r^2*(r - s)*s*(-1 + r^3*s) * Derivative[0, 1][QPochhammer][r/s, r^2*s^2]/ QPochhammer[r/s, r^2*s^2] + 1/s*k*(r - s)*(-1 + r^3*s) * (1/ Log[r^2*s^2]*(-2*QPolyGamma[0, 1, r^2*s^2] + QPolyGamma[0, Log[1/(r^3*s)] / Log[r^2*s^2], r^2*s^2] + QPolyGamma[0, Log[r/s] / Log[r^2*s^2], r^2*s^2]) + 2*r^2*s^2 * Derivative[0, 1][QPochhammer][r^2*s^2, r^2*s^2] / QPochhammer[r^2*s^2, r^2*s^2])) == 0} /. k -> 2, {r, 1/75}, {s, 2}, WorkingPrecision -> 70] (* Vaclav Kotesovec, Jan 18 2024 *)
  • PARI
    {a(n) = my(A=[2],M); for(i=1,n, A = concat(A,0); M = ceil(sqrt(n+1));
A[#A] = -polcoeff( sum(m=-M,M, (-x)^(m^2)*Ser(A)^((m-1)^2)), #A-1)); H=A; A[n+1]}
for(n=0,20,print1(a(n),", "))

Formula

G.f. A(x) satisfies:
(1) 2 = Sum_{n=-oo..+oo} (-x)^(n^2) * A(x)^((n+1)^2).
(2) 2 = A(x) * Product_{n>=1} (1 - x^(2*n)*A(x)^(2*n)) * (1 - x^(2*n-1)*A(x)^(2*n+1)) * (1 - x^(2*n-1)*A(x)^(2*n-3)), by the Jacobi triple product identity.
(3) 2 = (-x) * Product_{n>=1} (1 - x^(2*n)*A(x)^(2*n)) * (1 - x^(2*n+1)*A(x)^(2*n-1)) * (1 - x^(2*n-3)*A(x)^(2*n-1)), by the Jacobi triple product identity.
a(n) ~ c * d^n / n^(3/2), where d = 77.309779325704779292317107617559471210592218708634855530355675234... and c = 0.31219183409397424726366930735250286274022579073644627976468... - Vaclav Kotesovec, Mar 19 2023

A356503 G.f. A(x) satisfies: 3 = Sum_{n=-oo..+oo} (-x)^(n^2) * A(x)^((n-1)^2).

Original entry on oeis.org

3, 82, 8856, 1319544, 227536218, 42679033812, 8455886664768, 1741107313315440, 368888770098828828, 79897573332771325074, 17610753240158104125072, 3937441977622780631428392, 890818276864624495645873656, 203562312272030478854160019188, 46914726894168080421554447339136
Offset: 0

Views

Author

Paul D. Hanna, Aug 09 2022

Keywords

Examples

			G.f.: A(x) = 3 + 82*x + 8856*x^2 + 1319544*x^3 + 227536218*x^4 + 42679033812*x^5 + 8455886664768*x^6 + 1741107313315440*x^7 + 368888770098828828*x^8 + ...
such that A = A(x) satisfies
3 = ... + x^16*A^25 - x^9*A^16 + x^4*A^9 - x*A^4 + A - x + x^4*A - x^9*A^4 + x^16*A^9 - x^25*A^16 +- ... + (-x)^(n^2) * A(x,y)^((n-1)^2) + ...

Links

Crossrefs

Cf. A356500, A354248, A356502.

Programs

  • Mathematica
    (* Calculation of constant d: *) 1/r /. FindRoot[{k == r^4*s^2 * QPochhammer[1/(r^3*s), r^2*s^2] * QPochhammer[r/s, r^2*s^2] * QPochhammer[r^2*s^2, r^2*s^2]/((r - s)*(-1 + r^3*s)), 1/r^3*(k*(1 + r^4 - 2*r/s) + 2*r^6*s^3*QPochhammer[r/s, r^2*s^2] * QPochhammer[r^2*s^2, r^2*s^2]* Derivative[0, 1][QPochhammer][1/(r^3*s), r^2*s^2] + 2*k*r^2*(r - s)*s*(-1 + r^3*s) * Derivative[0, 1][QPochhammer][r/s, r^2*s^2]/ QPochhammer[r/s, r^2*s^2] + 1/s*k*(r - s)*(-1 + r^3*s) * (1/ Log[r^2*s^2]*(-2*QPolyGamma[0, 1, r^2*s^2] + QPolyGamma[0, Log[1/(r^3*s)] / Log[r^2*s^2], r^2*s^2] + QPolyGamma[0, Log[r/s] / Log[r^2*s^2], r^2*s^2]) + 2*r^2*s^2 * Derivative[0, 1][QPochhammer][r^2*s^2, r^2*s^2] / QPochhammer[r^2*s^2, r^2*s^2])) == 0} /. k -> 3, {r, 1/250}, {s, 2}, WorkingPrecision -> 70] (* Vaclav Kotesovec, Jan 18 2024 *)
  • PARI
    {a(n) = my(A=[3],M); for(i=1,n, A = concat(A,0); M = ceil(sqrt(n+1));
A[#A] = -polcoeff( sum(m=-M,M, (-x)^(m^2)*Ser(A)^((m-1)^2)), #A-1)); H=A; A[n+1]}
for(n=0,20,print1(a(n),", "))

Formula

G.f. A(x) satisfies:
(1) 3 = Sum_{n=-oo..+oo} (-x)^(n^2) * A(x)^((n+1)^2).
(2) 3 = A(x) * Product_{n>=1} (1 - x^(2*n)*A(x)^(2*n)) * (1 - x^(2*n-1)*A(x)^(2*n+1)) * (1 - x^(2*n-1)*A(x)^(2*n-3)), by the Jacobi triple product identity.
(3) 3 = (-x) * Product_{n>=1} (1 - x^(2*n)*A(x)^(2*n)) * (1 - x^(2*n+1)*A(x)^(2*n-1)) * (1 - x^(2*n-3)*A(x)^(2*n-1)), by the Jacobi triple product identity.
a(n) ~ c * d^n / n^(3/2), where d = 256.98186313678886207367060740797984009618789137465012036784040337275... and c = 0.462191657806050531963307936402124127070124175540050112607444087... - Vaclav Kotesovec, Mar 19 2023

A355872 G.f. A(x) satisfies: x = Sum_{n=-oo..+oo} (-x)^(n^2) * A(x)^((n-1)^2).

Original entry on oeis.org

2, 14, 434, 17662, 829314, 42293582, 2276970482, 127359871870, 7328894334338, 431089922960910, 25803242957983410, 1566580082112919422, 96239944539571023362, 5971465584401568096846, 373681955307631772312050, 23556948108319423559281918, 1494606013410312933197468930
Offset: 1

Views

Author

Paul D. Hanna, Aug 09 2022

Keywords

Comments

Conjecture: a(n) == 2 (mod 4) for n >= 1.
Conjecture: a(2*n-1) == 2 (mod 8) for n >= 1.
Conjecture: a(2*n) == 6 (mod 8) for n >= 1.
Equals the row sums of triangle A356501.

Examples

			G.f. A(x) = 2*x + 14*x^5 + 434*x^9 + 17662*x^13 + 829314*x^17 + 42293582*x^21 + 2276970482*x^25 + 127359871870*x^29 + 7328894334338*x^33 + 431089922960910*x^37 + ...
such that A = A(x) satisfies
x = ... + x^16*A^25 - x^9*A^16 + x^4*A^9 - x*A^4 + A - x + x^4*A - x^9*A^4 + x^16*A^9 - x^25*A^16 +- ... + (-x)^(n^2) * A(x,y)^((n-1)^2) + ...

Links

Crossrefs

Cf. A356500, A356501, A356502, A356503.

Programs

  • Mathematica
    (* Calculation of constant d: *) 1/r^4 /. FindRoot[{r*s^4*QPochhammer[1/(r*s^3), r^2*s^2] * QPochhammer[s/r, r^2*s^2]*QPochhammer[r^2*s^2, r^2*s^2] == (r - s)*(-1 + r*s^3), 1/s^3*(3*s + r*(-4 + r*s^3) + 2*r^2*(r - s)*s^2*(-1 + r*s^3)* Derivative[0, 1][QPochhammer][1/(r*s^3), r^2*s^2] / QPochhammer[1/(r*s^3), r^2*s^2] + 2*r^3*s^6*QPochhammer[1/(r*s^3), r^2*s^2] * QPochhammer[r^2*s^2, r^2*s^2] * Derivative[0, 1][QPochhammer][s/r, r^2*s^2] + (r - s)*(-1 + r*s^3)* (-(2*QPolyGamma[0, 1, r^2*s^2] - 3*QPolyGamma[0, Log[1/(r*s^3)]/Log[r^2*s^2], r^2*s^2] + QPolyGamma[0, Log[s/r]/Log[r^2*s^2], r^2*s^2]) / Log[r^2*s^2] + 2*r^2*s^2 * Derivative[0, 1][QPochhammer][r^2*s^2, r^2*s^2] /  QPochhammer[r^2*s^2, r^2*s^2])) == 0}, {r, 1/60}, {s, 2}, WorkingPrecision -> 70] (* Vaclav Kotesovec, Jan 18 2024 *)
  • PARI
    {a(n) = my(A=[0,2]); for(i=1,n, A=concat(A,[0,0,0,0]);
A[#A] = -polcoeff( sum(m=-#A,#A,(-x)^(m^2) * Ser(A)^((m-1)^2) ), #A-1)); A[4*n-2]}
for(n=1,20,print1(a(n),", "))

Formula

G.f. A(x) = Sum_{n>=1} a(n) * x^(4*n-3) satisfies:
(1) x = Sum_{n=-oo..+oo} (-x)^(n^2) * A(x)^((n+1)^2).
(2) x = A(x) * Product_{n>=1} (1 - x^(2*n)*A(x)^(2*n)) * (1 - x^(2*n-1)*A(x)^(2*n+1)) * (1 - x^(2*n-1)*A(x)^(2*n-3)), by the Jacobi triple product identity.
(3) -1 = Product_{n>=1} (1 - x^(2*n)*A(x)^(2*n)) * (1 - x^(2*n+1)*A(x)^(2*n-1)) * (1 - x^(2*n-3)*A(x)^(2*n-1)), by the Jacobi triple product identity.
a(n) ~ c * d^n / n^(3/2), where d = 69.7705416198088434764685861402300375255728007801297265... and c = 0.0044667602848752470638241640199049506066862963974858... - Vaclav Kotesovec, Mar 19 2023

A356501 Coefficients T(n,k) of x^(4*n+1-k)*y^k in A(x,y) for n >= 0, k = 0..3*n+1, where A(x,y) satisfies: y = Sum_{n=-oo..+oo} (-x)^(n^2) * A(x,y)^((n-1)^2), as an irregular triangle read by rows.

Original entry on oeis.org

1, 1, 0, 3, 6, 4, 1, 0, 9, 54, 120, 135, 84, 28, 4, 0, 22, 294, 1360, 3250, 4662, 4284, 2568, 981, 219, 22, 0, 51, 1260, 10120, 41405, 103020, 170324, 196172, 160965, 94390, 38896, 10764, 1807, 140, 0, 108, 4590, 58380, 368145, 1404102, 3587696, 6515712, 8715465, 8763645, 6684744, 3863496, 1670942, 525980, 114240, 15368, 969
Offset: 0

Views

Author

Paul D. Hanna, Aug 09 2022

Keywords

Comments

Row sums equal A355872.
Alternating row sums equals zero for all rows.
Rightmost border equals [x^n*y^(3*n+1)] A(x,y) = A002293(n) = binomial(4*n, n)/(3*n + 1).
This triangle may be formed from the nonzero antidiagonals of triangle A356500; see main entry A356500 for further formulas for the coefficients in g.f. A(x,y).

Examples

			G.f.: A(x,y) = y + x*(1 + y^4) + x^2*(4*y^3 + 4*y^7) + x^3*(6*y^2 + 28*y^6 + 22*y^10) + x^4*(3*y + 84*y^5 + 219*y^9 + 140*y^13) + x^5*(135*y^4 + 981*y^8 + 1807*y^12 + 969*y^16) + x^6*(120*y^3 + 2568*y^7 + 10764*y^11 + 15368*y^15 + 7084*y^19) + x^7*(54*y^2 + 4284*y^6 + 38896*y^10 + 114240*y^14 + 133266*y^18 + 53820*y^22) + x^8*(9*y + 4662*y^5 + 94390*y^9 + 525980*y^13 + 1187433*y^17 + 1171390*y^21 + 420732*y^25) + x^9*(3250*y^4 + 160965*y^8 + 1670942*y^12 + 6640711*y^16 + 12167001*y^20 + 10399545*y^24 + 3362260*y^28) + ...
such that A = A(x,y) satisfies
y = ... + x^16*A^25 - x^9*A^16 + x^4*A^9 - x*A^4 + A - x + x^4*A - x^9*A^4 + x^16*A^9 - x^25*A^16 +- ... + (-x)^(n^2) * A(x,y)^((n-1)^2) + ...
This triangle of coefficients of x^(4*n+1-k)*y^k in A(x,y) for n >= 0, k = 0..3*n+1, begins:
n = 0: [1, 1];
n = 1: [0, 3, 6, 4, 1];
n = 2: [0, 9, 54, 120, 135, 84, 28, 4];
n = 3: [0, 22, 294, 1360, 3250, 4662, 4284, 2568, 981, 219, 22];
n = 4: [0, 51, 1260, 10120, 41405, 103020, 170324, 196172, 160965, 94390, 38896, 10764, 1807, 140];
n = 5: [0, 108, 4590, 58380, 368145, 1404102, 3587696, 6515712, 8715465, 8763645, 6684744, 3863496, 1670942, 525980, 114240, 15368, 969];
n = 6: [0, 221, 14952, 282948, 2578147, 14039250, 51126740, 133101836, 258436719, 384735141, 446971668, 409367712, 296679006, 169713208, 75904032, 26050408, 6640711, 1187433, 133266, 7084];
...

Links

Crossrefs

Cf. A355872 (row sums), A356500 (main entry), A002293 (right border), A000716 (column 1).

Programs

  • PARI
    {T(n,k) = my(A=[y],M); for(i=1,4*n+1, A = concat(A,0); M = ceil(sqrt(4*n+1));
A[#A] = -polcoeff( sum(m=-M,M, (-x)^(m^2)*Ser(A)^((m-1)^2)), #A-1)); polcoeff(A[4*n+2-k],k,y)}
for(n=0,7, for(k=0,3*n+1, print1(T(n,k),", "));print(""))

Formula

G.f. A(x,y) = Sum_{n>=0} Sum_{k=0..3*n+1} T(n,k) * x^n * y^k satisfies:
(1) y = Sum_{n=-oo..+oo} (-x)^(n^2) * A(x,y)^((n+1)^2).
(2) y = A(x,y) * Product_{n>=1} (1 - x^(2*n)*A(x,y)^(2*n)) * (1 - x^(2*n-1)*A(x,y)^(2*n+1)) * (1 - x^(2*n-1)*A(x,y)^(2*n-3)), by the Jacobi triple product identity.
(3) y = (-x) * Product_{n>=1} (1 - x^(2*n)*A(x,y)^(2*n)) * (1 - x^(2*n+1)*A(x,y)^(2*n-1)) * (1 - x^(2*n-3)*A(x,y)^(2*n-1)), by the Jacobi triple product identity.
(4) y = A(x, F(x,y)) where F(x,y) = Sum_{n=-oo..+oo} (-x)^(n^2) * y^((n-1)^2).
(5) 1 = A(x, theta_4(x)) where theta_4(x) = 1 + 2*Sum_{n>=1} (-1)^n * x^(n^2) is a Jacobi theta function.
Showing 1-8 of 8 results.

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