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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A370020 Table in which the g.f. of row n, R(n,x), satisfies Sum_{k=-oo..+oo} (-1)^k * (x^k + n*R(n,x))^k = 1 + (n+2)*Sum_{k>=1} (-1)^k * x^(k^2), for n >= 1, as read by antidiagonals.

Original entry on oeis.org

1, 1, 1, 1, 2, 4, 1, 3, 7, 9, 1, 4, 12, 25, 22, 1, 5, 19, 53, 85, 63, 1, 6, 28, 99, 234, 301, 155, 1, 7, 39, 169, 529, 1041, 1086, 415, 1, 8, 52, 269, 1054, 2853, 4711, 3927, 1124, 1, 9, 67, 405, 1917, 6667, 15566, 21573, 14328, 2957, 1, 10, 84, 583, 3250, 13893, 42627, 85879, 99484, 52724, 8047, 1, 11, 103, 809, 5209, 26541, 101830, 275211, 477716, 461657, 194915, 21817
Offset: 1

Views

Author

Paul D. Hanna, Feb 09 2024

Keywords

Comments

A related function is theta_4(x) = 1 + 2*Sum_{n>=1} (-1)^n * x^(n^2).

Examples

			This table of coefficients T(n,k) of x^k in R(n,x), n >= 1, k >= 1, begins:
A370021: [1,  1,   4,    9,    22,     63,     155,      415, ...];
A370022: [1,  2,   7,   25,    85,    301,    1086,     3927, ...];
A370023: [1,  3,  12,   53,   234,   1041,    4711,    21573, ...];
A370024: [1,  4,  19,   99,   529,   2853,   15566,    85879, ...];
A370025: [1,  5,  28,  169,  1054,   6667,   42627,   275211, ...];
A370026: [1,  6,  39,  269,  1917,  13893,  101830,   753255, ...];
A370027: [1,  7,  52,  405,  3250,  26541,  219311,  1828657, ...];
A370028: [1,  8,  67,  583,  5209,  47341,  435366,  4039863, ...];
A370029: [1,  9,  84,  809,  7974,  79863,  809131,  8270199, ...];
A370042: [1, 10, 103, 1089, 11749, 128637, 1423982, 15898231, ...];
...
where the n-th row function R(n,x) satisfies
Sum_{k=-oo..+oo} (-1)^k * (x^k + n*R(n,x))^k = 1 + (n+2)*Sum_{k>=1} (-1)^k * x^(k^2).

Links

Crossrefs

Cf. A370021, A370022, A370023, A370024, A370025, A370026, A370027, A370028, A370029, A370042.
Cf. A370040, A370030 (dual table).

Programs

  • PARI
    {T(n,k) = my(A=[0,1]); for(i=0,k, A = concat(A,0);
A[#A] = polcoeff( sum(m=-#A,#A, (-1)^m * (x^m + n*Ser(A))^m ) - 1 - (n+2)*sum(m=1,#A, (-1)^m * x^(m^2) ), #A-1)/n ); A[k+1]}
for(n=1,12, for(k=1,10, print1(T(n,k),", "));print(""))

Formula

The n-th row g.f. R(n,x) = Sum_{k>=1} T(n,k)*x^k satisfies the following formulas.
(1) Sum_{k=-oo..+oo} (-1)^k * (x^k + n*R(n,x))^k = 1 + (n+2)*Sum_{k>=1} (-1)^k * x^(k^2).
(2) Sum_{k=-oo..+oo} (-1)^k * x^k * (x^k + n*R(n,x))^(k-1) = 1 + (n+2)*Sum_{k>=1} (-1)^k * x^(k^2).
(3) Sum_{k=-oo..+oo} (-1)^k * x^k * (x^k + n*R(n,x))^k = 0.
(4) Sum_{k=-oo..+oo} (-1)^k * x^(k^2) / (1 + n*R(n,x)*x^k)^k = 1 + (n+2)*Sum_{k>=1} (-1)^k * x^(k^2).
(5) Sum_{k=-oo..+oo} (-1)^k * x^(k^2) / (1 + n*R(n,x)*x^k)^(k+1) = 1 + (n+2)*Sum_{k>=1} (-1)^k * x^(k^2).
(6) Sum_{k=-oo..+oo} (-1)^k * x^(k*(k+1)) / (1 + n*R(n,x)*x^k)^(k+1) = 0.

A370021 Expansion of g.f. A(x) satisfying Sum_{n=-oo..+oo} (-1)^n * (x^n + A(x))^n = 1 + 3*Sum_{n>=1} (-1)^n * x^(n^2).

Original entry on oeis.org

1, 1, 4, 9, 22, 63, 155, 415, 1124, 2957, 8047, 21817, 59048, 161870, 442675, 1214563, 3348145, 9228858, 25514319, 70682731, 195993889, 544578231, 1515027660, 4219560585, 11768353857, 32853953466, 91812137378, 256831830373, 719046731299, 2014808160498, 5650037329385
Offset: 1

Views

Author

Paul D. Hanna, Feb 09 2024

Keywords

Comments

A related function is theta_4(x) = 1 + 2*Sum_{n>=1} (-1)^n * x^(n^2).
a(n+1)/a(n) tends to 2.874307... - Vaclav Kotesovec, Feb 11 2024

Examples

			G.f.: A(x) = x + x^2 + 4*x^3 + 9*x^4 + 22*x^5 + 63*x^6 + 155*x^7 + 415*x^8 + 1124*x^9 + 2957*x^10 + 8047*x^11 + 21817*x^12 + ...
where
Sum_{n=-oo..+oo} (-1)^n * (x^n + A(x))^n = 1 - 3*x + 3*x^4 - 3*x^9 + 3*x^16 - 3*x^25 + 3*x^36 - 3*x^49 +- ...
SPECIAL VALUES.
(V.1) Let A = A(exp(-Pi)) = 0.04543932020196352081239499480519595850147996376296857684...
then Sum_{n=-oo..+oo} (-1)^n * (exp(-n*Pi) + A)^n = (3*(Pi/2)^(1/4)/gamma(3/4) - 1)/2 = 0.87036870723417523211086...
(V.2) Let A = A(exp(-2*Pi)) = 0.00187095623366907901234297087932572258706353074482100743...
then Sum_{n=-oo..+oo} (-1)^n * (exp(-2*n*Pi) + A)^n = (3*2^(1/8)*(Pi/2)^(1/4)/gamma(3/4) - 1)/2 = 0.99626511456090713578995...
(V.3) Let A = A(-exp(-Pi)) = -0.04164083178192506029717066967023726841141127226704810579...
then Sum_{n=-oo..+oo} (-1)^n * (exp(-n*Pi) + A)^n = (3*Pi^(1/4)/gamma(3/4) - 1)/2 = 1.12965221681996202186297...
(V.4) Let A = A(-exp(-2*Pi)) = -0.00186398133004329627873834535037664668964585574963215266...
then Sum_{n=-oo..+oo} (-1)^n * (exp(-2*n*Pi) + A)^n = (3*sqrt(2 + sqrt(2))/2 * Pi^(1/4)/gamma(3/4) - 1)/2 = 1.00560232823160863657151...

Links

Crossrefs

Cf. A370020, A370022, A370023, A370024, A370025, A370026, A370027, A370028, A370029.

Programs

  • PARI
    {a(n) = my(A=[0,1]); for(i=0,n, A = concat(A,0);
A[#A] = polcoeff( sum(m=-#A,#A, (-1)^m * (x^m + 1*Ser(A))^m ) - 1 - 3*sum(m=1,#A, (-1)^m * x^(m^2) ), #A-1) ); A[n+1]}
for(n=1,30, print1(a(n),", "))

Formula

G.f. A(x) = Sum_{n>=1} a(n)*x^n satisfies the following formulas.
(1) Sum_{n=-oo..+oo} (-1)^n * (x^n + A(x))^n = 1 + 3*Sum_{n>=1} (-1)^n * x^(n^2).
(2) Sum_{n=-oo..+oo} (-1)^n * x^n * (x^n + A(x))^(n-1) = 1 + 3*Sum_{n>=1} (-1)^n * x^(n^2).
(3) Sum_{n=-oo..+oo} (-1)^n * x^n * (x^n + A(x))^n = 0.
(4) Sum_{n=-oo..+oo} (-1)^n * x^(n^2) / (1 + A(x)*x^n)^n = 1 + 3*Sum_{n>=1} (-1)^n * x^(n^2).
(5) Sum_{n=-oo..+oo} (-1)^n * x^(n^2) / (1 + A(x)*x^n)^(n+1) = 1 + 3*Sum_{n>=1} (-1)^n * x^(n^2).
(6) Sum_{n=-oo..+oo} (-1)^n * x^(n*(n+1)) / (1 + A(x)*x^n)^(n+1) = 0.

A370022 Expansion of g.f. A(x) satisfying Sum_{n=-oo..+oo} (-1)^n * (x^n + 2*A(x))^n = 1 + 4*Sum_{n>=1} (-1)^n * x^(n^2).

Original entry on oeis.org

1, 2, 7, 25, 85, 301, 1086, 3927, 14328, 52724, 194915, 723845, 2699878, 10104968, 37933855, 142795810, 538829973, 2037596590, 7720231359, 29302685197, 111398230285, 424115408181, 1616860117052, 6171586558551, 23583939930835, 90218328876825, 345461395176495, 1324041033133129
Offset: 1

Views

Author

Paul D. Hanna, Feb 09 2024

Keywords

Comments

A related function is theta_4(x) = 1 + 2*Sum_{n>=1} (-1)^n * x^(n^2).

Examples

			G.f.: A(x) = x + 2*x^2 + 7*x^3 + 25*x^4 + 85*x^5 + 301*x^6 + 1086*x^7 + 3927*x^8 + 14328*x^9 + 52724*x^10 + 194915*x^11 + 723845*x^12 + ...
where
Sum_{n=-oo..+oo} (-1)^n * (x^n + 2*A(x))^n = 1 - 4*x + 4*x^4 - 4*x^9 + 4*x^16 - 4*x^25 + 4*x^36 - 4*x^49 +- ...
SPECIAL VALUES.
(V.1) Let A = A(exp(-Pi)) = 0.04761601613534030259384050896565071457116692089742172541...
then Sum_{n=-oo..+oo} (-1)^n * (exp(-n*Pi) + 2*A)^n = 2*(Pi/2)^(1/4)/gamma(3/4) - 1 = 0.82715827631223364281448...
(V.2) Let A = A(exp(-2*Pi)) = 0.00187446330928756547025110339586987296984387228299321603...
then Sum_{n=-oo..+oo} (-1)^n * (exp(-2*n*Pi) + 2*A)^n = 2*2^(1/8)*(Pi/2)^(1/4)/gamma(3/4) - 1 = 0.99253022912181427157991...
(V.3) Let A = A(-exp(-Pi)) = -0.03996785964385216049635981950386915887875531406265280233...
then Sum_{n=-oo..+oo} (-1)^n * ((-1)^n*exp(-n*Pi) + 2*A)^n = 2*Pi^(1/4)/gamma(3/4) - 1 = 1.1728696224266160291506...
(V.4) Let A = A(-exp(-2*Pi)) = -0.00186051333175936112600864666861119312780357024086759004...
then Sum_{n=-oo..+oo} (-1)^n * ((-1)^n*exp(-2*n*Pi) + 2*A)^n = 2*sqrt(2 + sqrt(2))/2 * Pi^(1/4)/gamma(3/4) - 1 = 1.007469770975478182...

Links

Crossrefs

Cf. A370020, A370021, A370023, A370024, A370025, A370026, A370027, A370028, A370029, A370042.

Programs

  • PARI
    {a(n) = my(A=[0,1]); for(i=0,n, A = concat(A,0);
A[#A] = polcoeff( sum(m=-#A,#A, (-1)^m * (x^m + 2*Ser(A))^m ) - 1 - 4*sum(m=1,#A, (-1)^m * x^(m^2) ), #A-1)/2 ); A[n+1]}
for(n=1,30, print1(a(n),", "))

Formula

G.f. A(x) = Sum_{n>=1} a(n)*x^n satisfies the following formulas.
(1) Sum_{n=-oo..+oo} (-1)^n * (x^n + 2*A(x))^n = 1 + 4*Sum_{n>=1} (-1)^n * x^(n^2).
(2) Sum_{n=-oo..+oo} (-1)^n * x^n * (x^n + 2*A(x))^(n-1) = 1 + 4*Sum_{n>=1} (-1)^n * x^(n^2).
(3) Sum_{n=-oo..+oo} (-1)^n * x^n * (x^n + 2*A(x))^n = 0.
(4) Sum_{n=-oo..+oo} (-1)^n * x^(n^2) / (1 + 2*A(x)*x^n)^n = 1 + 4*Sum_{n>=1} (-1)^n * x^(n^2).
(5) Sum_{n=-oo..+oo} (-1)^n * x^(n^2) / (1 + 2*A(x)*x^n)^(n+1) = 1 + 4*Sum_{n>=1} (-1)^n * x^(n^2).
(6) Sum_{n=-oo..+oo} (-1)^n * x^(n*(n+1)) / (1 + 2*A(x)*x^n)^(n+1) = 0.

A370023 Expansion of g.f. A(x) satisfying Sum_{n=-oo..+oo} (-1)^n * (x^n + 3*A(x))^n = 1 + 5*Sum_{n>=1} (-1)^n * x^(n^2).

Original entry on oeis.org

1, 3, 12, 53, 234, 1041, 4711, 21573, 99484, 461657, 2154591, 10102701, 47555840, 224624016, 1064183887, 5055060411, 24068888061, 114841741098, 548992775523, 2628924592737, 12608597616161, 60558351876803, 291238387762452, 1402314223189959, 6759651098793285, 32617445956236720
Offset: 1

Views

Author

Paul D. Hanna, Feb 09 2024

Keywords

Comments

A related function is theta_4(x) = 1 + 2*Sum_{n>=1} (-1)^n * x^(n^2).

Examples

			G.f.: A(x) = x + 3*x^2 + 12*x^3 + 53*x^4 + 234*x^5 + 1041*x^6 + 4711*x^7 + 21573*x^8 + 99484*x^9 + 461657*x^10 + 2154591*x^11 + 10102701*x^12 + ...
where
Sum_{n=-oo..+oo} (-1)^n * (x^n + 3*A(x))^n = 1 - 5*x + 5*x^4 - 5*x^9 + 5*x^16 - 5*x^25 + 5*x^36 - 5*x^49 +- ...
SPECIAL VALUES.
(V.1) Let A = A(exp(-Pi)) = 0.05001316702398359971645418498866690386932728399152644693...
then Sum_{n=-oo..+oo} (-1)^n * (exp(-n*Pi) + 3*A)^n = (5*(Pi/2)^(1/4)/gamma(3/4) - 3)/2 = 0.78394784539029205351810...
(V.2) Let A = A(exp(-2*Pi)) = 0.001877983557643657576778844718492775838546798118866577860...
then Sum_{n=-oo..+oo} (-1)^n * (exp(-2*n*Pi) + 3*A)^n = (5*2^(1/8)*(Pi/2)^(1/4)/gamma(3/4) - 3)/2 = 0.990662786402267839474...
(V.3) Let A = A(-exp(-Pi)) = -0.03842474691590612761867206263978602696713545771404819339...
then Sum_{n=-oo..+oo} (-1)^n * ((-1)^n*exp(-n*Pi) + 3*A)^n = (5*Pi^(1/4)/gamma(3/4) - 3)/2 = 1.216087028033270036438...
(V.4) Let A = A(-exp(-2*Pi)) = -0.001857058214293085256892081751882664927312970576990961749...
then Sum_{n=-oo..+oo} (-1)^n * ((-1)^n*exp(-2*n*Pi) + 3*A)^n = (5*sqrt(2 + sqrt(2))/2 * Pi^(1/4)/gamma(3/4) - 3)/2 = 1.009337213719347727619...

Links

Crossrefs

Cf. A370020, A370021, A370022, A370024, A370025, A370026, A370027, A370028, A370029, A370042.

Programs

  • PARI
    {a(n) = my(A=[0,1]); for(i=1,n, A = concat(A,0);
A[#A] = polcoeff( sum(m=-#A,#A, (-1)^m * (x^m + 3*Ser(A))^m ) - 1 - 5*sum(m=1,#A, (-1)^m * x^(m^2) ), #A-1)/3 ); A[n+1]}
for(n=1,30, print1(a(n),", "))

Formula

G.f. A(x) = Sum_{n>=1} a(n)*x^n satisfies the following formulas.
(1) Sum_{n=-oo..+oo} (-1)^n * (x^n + 3*A(x))^n = 1 + 5*Sum_{n>=1} (-1)^n * x^(n^2).
(2) Sum_{n=-oo..+oo} (-1)^n * x^n * (x^n + 3*A(x))^(n-1) = 1 + 5*Sum_{n>=1} (-1)^n * x^(n^2).
(3) Sum_{n=-oo..+oo} (-1)^n * x^n * (x^n + 3*A(x))^n = 0.
(4) Sum_{n=-oo..+oo} (-1)^n * x^(n^2) / (1 + 3*A(x)*x^n)^n = 1 + 5*Sum_{n>=1} (-1)^n * x^(n^2).
(5) Sum_{n=-oo..+oo} (-1)^n * x^(n^2) / (1 + 3*A(x)*x^n)^(n+1) = 1 + 5*Sum_{n>=1} (-1)^n * x^(n^2).
(6) Sum_{n=-oo..+oo} (-1)^n * x^(n*(n+1)) / (1 + 3*A(x)*x^n)^(n+1) = 0.

A370024 Expansion of g.f. A(x) satisfying Sum_{n=-oo..+oo} (-1)^n * (x^n + 4*A(x))^n = 1 + 6*Sum_{n>=1} (-1)^n * x^(n^2).

Original entry on oeis.org

1, 4, 19, 99, 529, 2853, 15566, 85879, 477716, 2674070, 15047671, 85063429, 482733230, 2748703604, 15697194139, 89875431754, 515774659357, 2966016776556, 17088046518051, 98614323921685, 569967829487533, 3298876334401503, 19117753534875276, 110922240116613681, 644276475406441599
Offset: 1

Views

Author

Paul D. Hanna, Feb 09 2024

Keywords

Comments

A related function is theta_4(x) = 1 + 2*Sum_{n>=1} (-1)^n * x^(n^2).

Examples

			G.f.: A(x) = x + 4*x^2 + 19*x^3 + 99*x^4 + 529*x^5 + 2853*x^6 + 15566*x^7 + 85879*x^8 + 477716*x^9 + 2674070*x^10 + 15047671*x^11 + 85063429*x^12 + ...
where
Sum_{n=-oo..+oo} (-1)^n * (x^n + 4*A(x))^n = 1 - 6*x + 6*x^4 - 6*x^9 + 6*x^16 - 6*x^25 + 6*x^36 - 6*x^49 +- ...
SPECIAL VALUES.
(V.1) Let A = A(exp(-Pi)) = 0.05266628449954901094912490050067062239110765179054552678...
then Sum_{n=-oo..+oo} (-1)^n * (exp(-n*Pi) + 4*A)^n = 3*(Pi/2)^(1/4)/gamma(3/4) - 2 = 0.74073741446835046422172...
(V.2) Let A = A(exp(-2*Pi)) = 0.001881517053093894919707587041659521876650213322334450878...
then Sum_{n=-oo..+oo} (-1)^n * (exp(-2*n*Pi) + 4*A)^n = 3*2^(1/8)*(Pi/2)^(1/4)/gamma(3/4) - 2 = 0.988795343682721407369...
(V.3) Let A = A(-exp(-Pi)) = -0.03699687105031477666227946508842289849689211763245984347...
then Sum_{n=-oo..+oo} (-1)^n * ((-1)^n*exp(-n*Pi) + 4*A)^n = 3*Pi^(1/4)/gamma(3/4) - 2 = 1.259304433639924043725...
(V.4) Let A = A(-exp(-2*Pi)) = -0.0018536159060139689998658447922411419770684746756327600438...
then Sum_{n=-oo..+oo} (-1)^n * ((-1)^n*exp(-2*n*Pi) + 4*A)^n = 3*sqrt(2 + sqrt(2))/2 * Pi^(1/4)/gamma(3/4) - 2 = 1.011204656463217273143...

Links

Crossrefs

Cf. A370020, A370021, A370022, A370023, A370025, A370026, A370027, A370028, A370029, A370042.

Programs

  • PARI
    {a(n) = my(A=[0,1]); for(i=1,n, A = concat(A,0);
A[#A] = polcoeff( sum(m=-#A,#A, (-1)^m * (x^m + 4*Ser(A))^m ) - 1 - 6*sum(m=1,#A, (-1)^m * x^(m^2) ), #A-1)/4 ); A[n+1]}
for(n=1,30, print1(a(n),", "))

Formula

G.f. A(x) = Sum_{n>=1} a(n)*x^n satisfies the following formulas.
(1) Sum_{n=-oo..+oo} (-1)^n * (x^n + 4*A(x))^n = 1 + 6*Sum_{n>=1} (-1)^n * x^(n^2).
(2) Sum_{n=-oo..+oo} (-1)^n * x^n * (x^n + 4*A(x))^(n-1) = 1 + 6*Sum_{n>=1} (-1)^n * x^(n^2).
(3) Sum_{n=-oo..+oo} (-1)^n * x^n * (x^n + 4*A(x))^n = 0.
(4) Sum_{n=-oo..+oo} (-1)^n * x^(n^2) / (1 + 4*A(x)*x^n)^n = 1 + 6*Sum_{n>=1} (-1)^n * x^(n^2).
(5) Sum_{n=-oo..+oo} (-1)^n * x^(n^2) / (1 + 4*A(x)*x^n)^(n+1) = 1 + 6*Sum_{n>=1} (-1)^n * x^(n^2).
(6) Sum_{n=-oo..+oo} (-1)^n * x^(n*(n+1)) / (1 + 4*A(x)*x^n)^(n+1) = 0.

A370026 Expansion of g.f. A(x) satisfying Sum_{n=-oo..+oo} (-1)^n * (x^n + 6*A(x))^n = 1 + 8*Sum_{n>=1} (-1)^n * x^(n^2).

Original entry on oeis.org

1, 6, 39, 269, 1917, 13893, 101830, 753255, 5614504, 42110432, 317474187, 2403893757, 18270065438, 139305459960, 1065183756535, 8165168139498, 62729216570805, 482878316552298, 3723769699813119, 28762830132956421, 222495155932381229, 1723432870654770161, 13366099075223254740
Offset: 1

Views

Author

Paul D. Hanna, Feb 09 2024

Keywords

Comments

A related function is theta_4(x) = 1 + 2*Sum_{n>=1} (-1)^n * x^(n^2).

Examples

			G.f.: A(x) = x + 6*x^2 + 39*x^3 + 269*x^4 + 1917*x^5 + 13893*x^6 + 101830*x^7 + 753255*x^8 + 5614504*x^9 + 42110432*x^10 + 317474187*x^11 + ...
where
Sum_{n=-oo..+oo} (-1)^n * (x^n + 6*A(x))^n = 1 - 8*x + 8*x^4 - 8*x^9 + 8*x^16 - 8*x^25 + 8*x^36 - 8*x^49 +- ...
SPECIAL VALUES.
(V.1) Let A = A(exp(-Pi)) = 0.05892551210473733684254468528377030200762221986684224912...
then Sum_{n=-oo..+oo} (-1)^n * (exp(-n*Pi) + 6*A)^n = 4*(Pi/2)^(1/4)/gamma(3/4) - 3 = 0.65431655262446728562897...
(V.2) Let A = A(exp(-2*Pi)) = 0.001888624085511713374935799800784148455986111369097248489...
then Sum_{n=-oo..+oo} (-1)^n * (exp(-2*n*Pi) + 6*A)^n = 4*2^(1/8)*(Pi/2)^(1/4)/gamma(3/4) - 3 = 0.985060458243628543159...
(V.3) Let A = A(-exp(-Pi)) = -0.03443859231795915470687740421610270983167641847531807729...
then Sum_{n=-oo..+oo} (-1)^n * ((-1)^n*exp(-n*Pi) + 6*A)^n = 4*Pi^(1/4)/gamma(3/4) - 3 = 1.3457392448532320583012...
(V.4) Let A = A(-exp(-2*Pi)) = -0.001846769433141026637620872576636896819075507182864480219...
then Sum_{n=-oo..+oo} (-1)^n * ((-1)^n*exp(-2*n*Pi) + 6*A)^n = 4*sqrt(2 + sqrt(2))/2 * Pi^(1/4)/gamma(3/4) - 3 = 1.01493954195095636419...

Links

Crossrefs

Cf. A370020, A370021, A370022, A370023, A370024, A370025, A370027, A370028, A370029, A370042.

Programs

  • PARI
    {a(n) = my(A=[0,1]); for(i=1,n, A = concat(A,0);
A[#A] = polcoeff( sum(m=-#A,#A, (-1)^m * (x^m + 6*Ser(A))^m ) - 1 - 8*sum(m=1,#A, (-1)^m * x^(m^2) ), #A-1)/6 ); A[n+1]}
for(n=1,30, print1(a(n),", "))

Formula

G.f. A(x) = Sum_{n>=1} a(n)*x^n satisfies the following formulas.
(1) Sum_{n=-oo..+oo} (-1)^n * (x^n + 6*A(x))^n = 1 + 8*Sum_{n>=1} (-1)^n * x^(n^2).
(2) Sum_{n=-oo..+oo} (-1)^n * x^n * (x^n + 6*A(x))^(n-1) = 1 + 8*Sum_{n>=1} (-1)^n * x^(n^2).
(3) Sum_{n=-oo..+oo} (-1)^n * x^n * (x^n + 6*A(x))^n = 0.
(4) Sum_{n=-oo..+oo} (-1)^n * x^(n^2) / (1 + 6*A(x)*x^n)^n = 1 + 8*Sum_{n>=1} (-1)^n * x^(n^2).
(5) Sum_{n=-oo..+oo} (-1)^n * x^(n^2) / (1 + 6*A(x)*x^n)^(n+1) = 1 + 8*Sum_{n>=1} (-1)^n * x^(n^2).
(6) Sum_{n=-oo..+oo} (-1)^n * x^(n*(n+1)) / (1 + 6*A(x)*x^n)^(n+1) = 0.

A370027 Expansion of g.f. A(x) satisfying Sum_{n=-oo..+oo} (-1)^n * (x^n + 7*A(x))^n = 1 + 9*Sum_{n>=1} (-1)^n * x^(n^2).

Original entry on oeis.org

1, 7, 52, 405, 3250, 26541, 219311, 1828657, 15360068, 129802889, 1102476535, 9403920685, 80507808128, 691425600548, 5954703569335, 51409228587355, 444806083780093, 3856115167020090, 33488422645226379, 291294693699275917, 2537471770952346625, 22133307405655321131
Offset: 1

Views

Author

Paul D. Hanna, Feb 09 2024

Keywords

Comments

A related function is theta_4(x) = 1 + 2*Sum_{n>=1} (-1)^n * x^(n^2).

Examples

			G.f.: A(x) = x + 7*x^2 + 52*x^3 + 405*x^4 + 3250*x^5 + 26541*x^6 + 219311*x^7 + 1828657*x^8 + 15360068*x^9 + 129802889*x^10 + 1102476535*x^11 + 9403920685*x^12 + ...
where
Sum_{n=-oo..+oo} (-1)^n * (x^n + 7*A(x))^n = 1 - 9*x + 9*x^4 - 9*x^9 + 9*x^16 - 9*x^25 + 9*x^36 - 9*x^49 +- ...
SPECIAL VALUES.
(V.1) Let A = A(exp(-Pi)) = 0.06265408791983395104830182276472061307372169283289177444...
then Sum_{n=-oo..+oo} (-1)^n * (exp(-n*Pi) + 7*A)^n = (9*(Pi/2)^(1/4)/gamma(3/4) - 7)/2 = 0.6111061217025256963...
(V.2) Let A = A(exp(-2*Pi)) = 0.001892197774017068345453024031418945825808997896316975979...
then Sum_{n=-oo..+oo} (-1)^n * (exp(-2*n*Pi) + 7*A)^n = (9*2^(1/8)*(Pi/2)^(1/4)/gamma(3/4) - 7)/2 = 0.98319301552408211105...
(V.3) Let A = A(-exp(-Pi)) = -0.03328815108533045197898037729675109506494860109014140530...
then Sum_{n=-oo..+oo} (-1)^n * ((-1)^n*exp(-n*Pi) + 7*A)^n = (9*Pi^(1/4)/gamma(3/4) - 7)/2 = 1.388956650459886065588...
(V.4) Let A = A(-exp(-2*Pi)) = -0.001843365127917378852723125074532830028319143070315792225...
then Sum_{n=-oo..+oo} (-1)^n * ((-1)^n*exp(-2*n*Pi) + 7*A)^n = (9*sqrt(2 + sqrt(2))/2 * Pi^(1/4)/gamma(3/4) - 7)/2 = 1.0168069846948259097...

Links

Crossrefs

Cf. A370020, A370021, A370022, A370023, A370024, A370025, A370026, A370028, A370029.

Programs

  • PARI
    {a(n) = my(A=[0,1]); for(i=1,n, A = concat(A,0);
A[#A] = polcoeff( sum(m=-#A,#A, (-1)^m * (x^m + 7*Ser(A))^m ) - 1 - 9*sum(m=1,#A, (-1)^m * x^(m^2) ), #A-1)/7 ); A[n+1]}
for(n=1,30, print1(a(n),", "))

Formula

G.f. A(x) = Sum_{n>=1} a(n)*x^n satisfies the following formulas.
(1) Sum_{n=-oo..+oo} (-1)^n * (x^n + 7*A(x))^n = 1 + 9*Sum_{n>=1} (-1)^n * x^(n^2).
(2) Sum_{n=-oo..+oo} (-1)^n * x^n * (x^n + 7*A(x))^(n-1) = 1 + 9*Sum_{n>=1} (-1)^n * x^(n^2).
(3) Sum_{n=-oo..+oo} (-1)^n * x^n * (x^n + 7*A(x))^n = 0.
(4) Sum_{n=-oo..+oo} (-1)^n * x^(n^2) / (1 + 7*A(x)*x^n)^n = 1 + 9*Sum_{n>=1} (-1)^n * x^(n^2).
(5) Sum_{n=-oo..+oo} (-1)^n * x^(n^2) / (1 + 7*A(x)*x^n)^(n+1) = 1 + 9*Sum_{n>=1} (-1)^n * x^(n^2).
(6) Sum_{n=-oo..+oo} (-1)^n * x^(n*(n+1)) / (1 + 7*A(x)*x^n)^(n+1) = 0.

A370028 Expansion of g.f. A(x) satisfying Sum_{n=-oo..+oo} (-1)^n * (x^n + 8*A(x))^n = 1 + 10*Sum_{n>=1} (-1)^n * x^(n^2).

Original entry on oeis.org

1, 8, 67, 583, 5209, 47341, 435366, 4039863, 37756884, 354968162, 3353718911, 31818650141, 302968462870, 2893794722996, 27715660576627, 266092098125266, 2560193682174621, 24680314094825608, 238332314224287603, 2305147105334586877, 22327315195346300461, 216542482388830668603
Offset: 1

Views

Author

Paul D. Hanna, Feb 09 2024

Keywords

Comments

A related function is theta_4(x) = 1 + 2*Sum_{n>=1} (-1)^n * x^(n^2).

Examples

			G.f.: A(x) = x + 8*x^2 + 67*x^3 + 583*x^4 + 5209*x^5 + 47341*x^6 + 435366*x^7 + 4039863*x^8 + 37756884*x^9 + 354968162*x^10 + 3353718911*x^11 + 31818650141*x^12 + ...
where
Sum_{n=-oo..+oo} (-1)^n * (x^n + 8*A(x))^n = 1 - 10*x + 10*x^4 - 10*x^9 + 10*x^16 - 10*x^25 + 10*x^36 - 10*x^49 +- ...
SPECIAL VALUES.
(V.1) Let A = A(exp(-Pi)) = 0.06689190492526765287210924306086051922855300119805422530...
then Sum_{n=-oo..+oo} (-1)^n * (exp(-n*Pi) + 8*A)^n = 5*(Pi/2)^(1/4)/gamma(3/4) - 4 = 0.5678956907805841070...
(V.2) Let A = A(exp(-2*Pi)) = 0.001892197774017068345453024031418945825808997896316975979...
then Sum_{n=-oo..+oo} (-1)^n * (exp(-2*n*Pi) + 8*A)^n = 5*2^(1/8)*(Pi/2)^(1/4)/gamma(3/4) - 4 = 0.98319301552408211105...
(V.3) Let A = A(-exp(-Pi)) = -0.03328815108533045197898037729675109506494860109014140530...
then Sum_{n=-oo..+oo} (-1)^n * ((-1)^n*exp(-n*Pi) + 8*A)^n = 5*Pi^(1/4)/gamma(3/4) - 4 = 1.432174056066540072876...
(V.4) Let A = A(-exp(-2*Pi)) = -0.001839973350611618077357159042562240768956638628903670470...
then Sum_{n=-oo..+oo} (-1)^n * ((-1)^n*exp(-2*n*Pi) + 8*A)^n = 5*sqrt(2 + sqrt(2))/2 * Pi^(1/4)/gamma(3/4) - 4 = 1.0186744274386954552...

Links

Crossrefs

Cf. A370020, A370021, A370022, A370023, A370024, A370025, A370026, A370027, A370029, A370042.

Programs

  • PARI
    {a(n) = my(A=[0,1]); for(i=1,n, A = concat(A,0);
A[#A] = polcoeff( sum(m=-#A,#A, (-1)^m * (x^m + 8*Ser(A))^m ) - 1 - 10*sum(m=1,#A, (-1)^m * x^(m^2) ), #A-1)/8 ); A[n+1]}
for(n=1,30, print1(a(n),", "))

Formula

G.f. A(x) = Sum_{n>=1} a(n)*x^n satisfies the following formulas.
(1) Sum_{n=-oo..+oo} (-1)^n * (x^n + 8*A(x))^n = 1 + 10*Sum_{n>=1} (-1)^n * x^(n^2).
(2) Sum_{n=-oo..+oo} (-1)^n * x^n * (x^n + 8*A(x))^(n-1) = 1 + 10*Sum_{n>=1} (-1)^n * x^(n^2).
(3) Sum_{n=-oo..+oo} (-1)^n * x^n * (x^n + 8*A(x))^n = 0.
(4) Sum_{n=-oo..+oo} (-1)^n * x^(n^2) / (1 + 8*A(x)*x^n)^n = 1 + 10*Sum_{n>=1} (-1)^n * x^(n^2).
(5) Sum_{n=-oo..+oo} (-1)^n * x^(n^2) / (1 + 8*A(x)*x^n)^(n+1) = 1 + 10*Sum_{n>=1} (-1)^n * x^(n^2).
(6) Sum_{n=-oo..+oo} (-1)^n * x^(n*(n+1)) / (1 + 8*A(x)*x^n)^(n+1) = 0.

A370029 Expansion of g.f. A(x) satisfying Sum_{n=-oo..+oo} (-1)^n * (x^n + 9*A(x))^n = 1 + 11*Sum_{n>=1} (-1)^n * x^(n^2).

Original entry on oeis.org

1, 9, 84, 809, 7974, 79863, 809131, 8270199, 85126516, 881290445, 9167900511, 95763822969, 1003839653480, 10554997636854, 111280621221379, 1176017223671139, 12454545436154097, 132149953604522106, 1404591515239624671, 14952277258870348035, 159396459604398283553
Offset: 1

Views

Author

Paul D. Hanna, Feb 09 2024

Keywords

Comments

A related function is theta_4(x) = 1 + 2*Sum_{n>=1} (-1)^n * x^(n^2).

Examples

			G.f.: A(x) = x + 9*x^2 + 84*x^3 + 809*x^4 + 7974*x^5 + 79863*x^6 + 809131*x^7 + 8270199*x^8 + 85126516*x^9 + 881290445*x^10 + 9167900511*x^11 + 95763822969*x^12 + ...
where
Sum_{n=-oo..+oo} (-1)^n * (x^n + 9*A(x))^n = 1 - 11*x + 11*x^4 - 11*x^9 + 11*x^16 - 11*x^25 + 11*x^36 - 11*x^49 +- ...
SPECIAL VALUES.
(V.1) Let A = A(exp(-Pi)) = 0.07175219834164736620386280600888962717215573957821859403...
then Sum_{n=-oo..+oo} (-1)^n * (exp(-n*Pi) + 9*A)^n = (11*(Pi/2)^(1/4)/gamma(3/4) - 9)/2 = 0.5246852598586425177...
(V.2) Let A = A(exp(-2*Pi)) = 0.001899385878782719362352219788087550672661478114904760835...
then Sum_{n=-oo..+oo} (-1)^n * (exp(-2*n*Pi) + 9*A)^n = (11*2^(1/8)*(Pi/2)^(1/4)/gamma(3/4) - 9)/2 = 0.97945813008498924684...
(V.3) Let A = A(-exp(-Pi)) = -0.03120408533767785789845054540220571847531668789278074466...
then Sum_{n=-oo..+oo} (-1)^n * ((-1)^n*exp(-n*Pi) + 9*A)^n = (11*Pi^(1/4)/gamma(3/4) - 9)/2 = 1.475391461673194080164...
(V.4) Let A = A(-exp(-2*Pi)) = -0.001836594032195533189068390983153367342311468510211476381...
then Sum_{n=-oo..+oo} (-1)^n * ((-1)^n*exp(-2*n*Pi) + 9*A)^n = (11*sqrt(2 + sqrt(2))/2 * Pi^(1/4)/gamma(3/4) - 9)/2 = 1.0205418701825650...

Links

Crossrefs

Cf. A370020, A370021, A370022, A370023, A370024, A370025, A370026, A370027, A370028, A370042.

Programs

  • PARI
    {a(n) = my(A=[0,1]); for(i=1,n, A = concat(A,0);
A[#A] = polcoeff( sum(m=-#A,#A, (-1)^m * (x^m + 9*Ser(A))^m ) - 1 - 11*sum(m=1,#A, (-1)^m * x^(m^2) ), #A-1)/9 ); A[n+1]}
for(n=1,30, print1(a(n),", "))

Formula

G.f. A(x) = Sum_{n>=1} a(n)*x^n satisfies the following formulas.
(1) Sum_{n=-oo..+oo} (-1)^n * (x^n + 9*A(x))^n = 1 + 11*Sum_{n>=1} (-1)^n * x^(n^2).
(2) Sum_{n=-oo..+oo} (-1)^n * x^n * (x^n + 9*A(x))^(n-1) = 1 + 11*Sum_{n>=1} (-1)^n * x^(n^2).
(3) Sum_{n=-oo..+oo} (-1)^n * x^n * (x^n + 9*A(x))^n = 0.
(4) Sum_{n=-oo..+oo} (-1)^n * x^(n^2) / (1 + 9*A(x)*x^n)^n = 1 + 11*Sum_{n>=1} (-1)^n * x^(n^2).
(5) Sum_{n=-oo..+oo} (-1)^n * x^(n^2) / (1 + 9*A(x)*x^n)^(n+1) = 1 + 11*Sum_{n>=1} (-1)^n * x^(n^2).
(6) Sum_{n=-oo..+oo} (-1)^n * x^(n*(n+1)) / (1 + 9*A(x)*x^n)^(n+1) = 0.

A370042 Expansion of g.f. A(x) satisfying Sum_{n=-oo..+oo} (-1)^n * (x^n + 10*A(x))^n = 1 + 12*Sum_{n>=1} (-1)^n * x^(n^2).

Original entry on oeis.org

1, 10, 103, 1089, 11749, 128637, 1423982, 15898231, 178717112, 2020360748, 22947819571, 261696375829, 2994717484790, 34373295184712, 395580223408591, 4563146810297938, 52747346257279381, 610871638149166758, 7086520419499114527, 82334442066436896541, 957935578573905521101
Offset: 1

Views

Author

Paul D. Hanna, Feb 08 2024

Keywords

Comments

A related function is theta_4(x) = 1 + 2*Sum_{n>=1} (-1)^n * x^(n^2).

Examples

			G.f.: A(x) = x + 10*x^2 + 103*x^3 + 1089*x^4 + 11749*x^5 + 128637*x^6 + 1423982*x^7 + 15898231*x^8 + 178717112*x^9 + 2020360748*x^10 + 22947819571*x^11 + ...
where
Sum_{n=-oo..+oo} (-1)^n * (x^n + 10*A(x))^n = 1 - 12*x + 12*x^4 - 12*x^9 + 12*x^16 - 12*x^25 + 12*x^36 - 12*x^49 +- ...
SPECIAL VALUES.
(V.1) Let A = A(exp(-Pi)) = 0.07738488286995169642543180751945321776018365032150702566...
then Sum_{n=-oo..+oo} (-1)^n * (exp(-n*Pi) + 10*A)^n = 6*(Pi/2)^(1/4)/gamma(3/4) - 5 = 0.4814748289367009284...
(V.2) Let A = A(exp(-2*Pi)) = 0.001903000450057888437867399675031908155434474357834107336...
then Sum_{n=-oo..+oo} (-1)^n * (exp(-2*n*Pi) + 10*A)^n = 6*2^(1/8)*(Pi/2)^(1/4)/gamma(3/4) - 5 = 0.9775906873654428147...
(V.3) Let A = A(-exp(-Pi)) = -0.03025721520362353256298796517975081121112509387406260314...
then Sum_{n=-oo..+oo} (-1)^n * ((-1)^n*exp(-n*Pi) + 10*A)^n = 6*Pi^(1/4)/gamma(3/4) - 5 = 1.518608867279848087...
(V.4) Let A = A(-exp(-2*Pi)) = -0.001833227104147111248063467420834469150941590720555554025...
then Sum_{n=-oo..+oo} (-1)^n * ((-1)^n*exp(-2*n*Pi) + 10*A)^n = 6*sqrt(2 + sqrt(2))/2 * Pi^(1/4)/gamma(3/4) - 5 = 1.0224093129264345...

Links

Crossrefs

Cf. A370020, A370021, A370022, A370023, A370024, A370025, A370026, A370027, A370028, A370029.

Programs

  • PARI
    {a(n) = my(A=[0,1]); for(i=1,n, A = concat(A,0);
A[#A] = polcoeff( sum(m=-#A,#A, (-1)^m * (x^m + 10*Ser(A))^m ) - 1 - 12*sum(m=1,#A, (-1)^m * x^(m^2) ), #A-1)/10 ); A[n+1]}
for(n=1,30, print1(a(n),", "))

Formula

G.f. A(x) = Sum_{n>=1} a(n)*x^n satisfies the following formulas.
(1) Sum_{n=-oo..+oo} (-1)^n * (x^n + 10*A(x))^n = 1 + 12*Sum_{n>=1} (-1)^n * x^(n^2).
(2) Sum_{n=-oo..+oo} (-1)^n * x^n * (x^n + 10*A(x))^(n-1) = 1 + 12*Sum_{n>=1} (-1)^n * x^(n^2).
(3) Sum_{n=-oo..+oo} (-1)^n * x^n * (x^n + 10*A(x))^n = 0.
(4) Sum_{n=-oo..+oo} (-1)^n * x^(n^2) / (1 + 10*A(x)*x^n)^n = 1 + 12*Sum_{n>=1} (-1)^n * x^(n^2).
(5) Sum_{n=-oo..+oo} (-1)^n * x^(n^2) / (1 + 10*A(x)*x^n)^(n+1) = 1 + 12*Sum_{n>=1} (-1)^n * x^(n^2).
(6) Sum_{n=-oo..+oo} (-1)^n * x^(n*(n+1)) / (1 + 10*A(x)*x^n)^(n+1) = 0.
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