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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A370039 Expansion of g.f. A(x) satisfying Sum_{n=-oo..+oo} (x^n - 9*A(x))^n = 1 - 7*Sum_{n>=1} x^(n^2).

Original entry on oeis.org

1, 9, 80, 703, 6130, 53351, 466315, 4118167, 36941188, 337853203, 3155619199, 30087573015, 292226014968, 2882482639376, 28783571541579, 290149337803965, 2945978857054165, 30080058358496842, 308542728377796463, 3177317808394936571, 32835881264222087409, 340467815173685043729
Offset: 1

Views

Author

Paul D. Hanna, Feb 10 2024

Keywords

Comments

A related function is theta_3(x) = 1 + 2*Sum_{n>=1} x^(n^2).

Examples

			G.f.: A(x) = x + 9*x^2 + 80*x^3 + 703*x^4 + 6130*x^5 + 53351*x^6 + 466315*x^7 + 4118167*x^8 + 36941188*x^9 + 337853203*x^10 + 3155619199*x^11 + ...
where
Sum_{n=-oo..+oo} (x^n - 8*A(x))^n = 1 - 7*x - 7*x^4 - 7*x^9 - 7*x^16 - 7*x^25 - 7*x^36 - 7*x^49 - ...
SPECIAL VALUES.
(V.1) Let A = A(exp(-Pi)) = 0.07041342765468695859173243504212855904085321490660808668...
then Sum_{n=-oo..+oo} (exp(-n*Pi) - 9*A)^n = (9 - 7*Pi^(1/4)/gamma(3/4))/2 = 0.69747816075342194898639...
(V.2) Let A = A(exp(-2*Pi)) = 0.001899358496977867055016493259704554658290299283307899768...
then Sum_{n=-oo..+oo} (exp(-2*n*Pi) - 9*A)^n = (9 - 7*sqrt(2 + sqrt(2))/2 * Pi^(1/4)/gamma(3/4))/2 = 0.98692790079291318133312...
(V.3) Let A = A(-exp(-Pi)) = -0.03108273985731889208644710399967055047528520340415555251...
then Sum_{n=-oo..+oo} ((-1)^n*exp(-n*Pi) - 9*A)^n = (9 - 7*(Pi/2)^(1/4)/gamma(3/4))/2 = 1.302473016453591125074...
(V.4) Let A = A(-exp(-2*Pi)) = -0.001836569230890760040434767580223720991124539653197115902...
then Sum_{n=-oo..+oo} ((-1)^n*exp(-2*n*Pi) - 9*A)^n = (9 - 7*2^(1/8)*(Pi/2)^(1/4)/gamma(3/4))/2 = 1.013072099036825024735...

Links

Crossrefs

Cf. A370041, A370030, A370031, A355868, A370033, A370034, A370035, A370036, A370037, A370038, A370043.

Programs

  • PARI
    {a(n) = my(A=[0,1]); for(i=1,n, A = concat(A,0);
A[#A] = polcoeff( sum(m=-#A,#A, (x^m - 9*Ser(A))^m ) - 1 + 7*sum(m=1,#A, 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} (x^n - 9*A(x))^n = 1 - 7*Sum_{n>=1} x^(n^2).
(2) Sum_{n=-oo..+oo} x^n * (x^n + 9*A(x))^(n-1) = 1 - 7*Sum_{n>=1} x^(n^2).
(3) Sum_{n=-oo..+oo} (-1)^n * x^n * (x^n - 9*A(x))^n = 0.
(4) Sum_{n=-oo..+oo} x^(n^2) / (1 - 9*x^n*A(x))^n = 1 - 7*Sum_{n>=1} x^(n^2).
(5) Sum_{n=-oo..+oo} x^(n^2) / (1 + 9*x^n*A(x))^(n+1) = 1 - 7*Sum_{n>=1} x^(n^2).
(6) Sum_{n=-oo..+oo} (-1)^n * x^(n*(n-1)) / (1 - 9*x^n*A(x))^n = 0.

A370043 Expansion of g.f. A(x) satisfying Sum_{n=-oo..+oo} (x^n - 10*A(x))^n = 1 - 8*Sum_{n>=1} x^(n^2).

Original entry on oeis.org

1, 10, 99, 971, 9461, 91959, 895518, 8775161, 86870264, 871650208, 8884142855, 92061370003, 969550433086, 10363557226896, 112215017274331, 1228207449471086, 13561137797537413, 150791851996365182, 1686274213530482843, 18945675318778308411, 213704510012147008821
Offset: 1

Views

Author

Paul D. Hanna, Feb 10 2024

Keywords

Comments

A related function is theta_3(x) = 1 + 2*Sum_{n>=1} x^(n^2).

Examples

			G.f.: A(x) = x + 10*x^2 + 99*x^3 + 971*x^4 + 9461*x^5 + 91959*x^6 + 895518*x^7 + 8775161*x^8 + 86870264*x^9 + 871650208*x^10 + 8884142855*x^11 + ...
where
Sum_{n=-oo..+oo} (x^n - 8*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)) = 0.07572861892245027340976642864415638836692678958408803376...
then Sum_{n=-oo..+oo} (exp(-n*Pi) - 10*A)^n = 5 - 4*Pi^(1/4)/gamma(3/4) = 0.6542607551467679416987...
(V.2) Let A = A(exp(-2*Pi)) = 0.001902972911784356118532074933211699956337964100195554269...
then Sum_{n=-oo..+oo} (exp(-2*n*Pi) - 10*A)^n = 5 - 4*sqrt(2 + sqrt(2))/2 * Pi^(1/4)/gamma(3/4) = 0.9850604580490436358...
(V.3) Let A = A(-exp(-Pi)) = -0.03014664142938059660934561948726688645121488051083843222...
then Sum_{n=-oo..+oo} ((-1)^n*exp(-n*Pi) - 10*A)^n = 5 - 4*(Pi/2)^(1/4)/gamma(3/4) = 1.34568344737553271437...
(V.4) Let A = A(-exp(-2*Pi)) = -0.001833202439114209450155973975718938793478260093149995057...
then Sum_{n=-oo..+oo} ((-1)^n*exp(-2*n*Pi) - 10*A)^n = 5 - 4*2^(1/8)*(Pi/2)^(1/4)/gamma(3/4) = 1.0149395417563714568...

Links

Crossrefs

Cf. A370041, A370030, A370031, A355868, A370033, A370034, A370035, A370036, A370037, A370038, A370039.

Programs

  • PARI
    {a(n) = my(A=[0,1]); for(i=1,n, A = concat(A,0);
A[#A] = polcoeff( sum(m=-#A,#A, (x^m - 10*Ser(A))^m ) - 1 + 8*sum(m=1,#A, 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} (x^n - 10*A(x))^n = 1 - 8*Sum_{n>=1} x^(n^2).
(2) Sum_{n=-oo..+oo} x^n * (x^n + 10*A(x))^(n-1) = 1 - 8*Sum_{n>=1} x^(n^2).
(3) Sum_{n=-oo..+oo} (-1)^n * x^n * (x^n - 10*A(x))^n = 0.
(4) Sum_{n=-oo..+oo} x^(n^2) / (1 - 10*x^n*A(x))^n = 1 - 8*Sum_{n>=1} x^(n^2).
(5) Sum_{n=-oo..+oo} x^(n^2) / (1 + 10*x^n*A(x))^(n+1) = 1 - 8*Sum_{n>=1} x^(n^2).
(6) Sum_{n=-oo..+oo} (-1)^n * x^(n*(n-1)) / (1 - 10*x^n*A(x))^n = 0.

A357232 a(n) = coefficient of x^n, n >= 0, in A(x) such that: 2 = Sum_{n=-oo..+oo} (-1)^n * x^n * (2*A(x) + x^n)^(2*n+1).

Original entry on oeis.org

1, 3, 25, 254, 2763, 32180, 393169, 4964017, 64254694, 848214039, 11375359344, 154547261539, 2122630191360, 29423373611509, 411105855956011, 5783709944279141, 81862107418919278, 1164873718427628846, 16654829725736560441, 239140138388082634266, 3446933945466334214525
Offset: 0

Views

Author

Paul D. Hanna, Oct 14 2022

Keywords

Comments

Compare to A355865.

Examples

			G.f.: A(x) = 1 + 3*x + 25*x^2 + 254*x^3 + 2763*x^4 + 32180*x^5 + 393169*x^6 + 4964017*x^7 + 64254694*x^8 + 848214039*x^9 + 11375359344*x^10 + ...

Links

Crossrefs

Cf. A355865, A355868, A357402.

Programs

  • PARI
    {a(n) = my(A=1); for(L=1,n, A = truncate(A);
A = A + 1 - (1/2)*sum(m=-L,L, (-1)^m * x^m * (2*A + x^m +x^2*O(x^(L+1)))^(2*m+1) ) ); polcoeff(A,n)}
for(n=0,30, print1(a(n),", "))

Formula

Generating function A(x) = Sum_{n>=0} a(n)*x^n satisfies the following formulas.
(1) 2 = Sum_{n=-oo..+oo} (-1)^n * x^n * (2*A(x) + x^n)^(2*n+1).
(2) 2 = Sum_{n=-oo..+oo} (-1)^n * x^(2*n*(n-1)) / (1 + 2*A(x)*x^n)^(2*n-1).

A369671 Expansion of g.f. A(x) satisfying Sum_{n=-oo..+oo} (x^n - 4*A(x))^n = theta_4(x).

Original entry on oeis.org

1, 4, 15, 52, 177, 664, 3038, 16268, 90660, 490456, 2541387, 12819184, 64665462, 333763444, 1776226471, 9670530120, 53128162973, 291546645940, 1592977754671, 8685610041084, 47462008167381, 260789472093044, 1442162566738036, 8016343531922084, 44697615509640615, 249596790724248848
Offset: 1

Views

Author

Paul D. Hanna, Feb 03 2024

Keywords

Comments

Note: theta_4(x) = Sum_{n=-oo..+oo} (-1)^n * x^(n^2) - see A002448.
Congruences:
(C.1) a(2*n) == 0 (mod 4) for n >= 1.
(C.2) a(n) == A369672(n) (mod 4) for n >= 1.
(C.3) a(2*n)/4 == -A369672(2*n)/4 (mod 4) for n >= 1.

Examples

			G.f.: A(x) = x + 4*x^2 + 15*x^3 + 52*x^4 + 177*x^5 + 664*x^6 + 3038*x^7 + 16268*x^8 + 90660*x^9 + 490456*x^10 + 2541387*x^11 + 12819184*x^12 + ...
where Sum_{n=-oo..+oo} (x^n - 4*A(x))^n = theta_4(x), and
theta_4(x) = 1 - 2*x + 2*x^4 - 2*x^9 + 2*x^16 + ... + (-1)^n*2*x^(n^2) + ...
RELATED SERIES.
When we break up the doubly infinite sum into the following parts
P = Sum_{n>=0} (x^n - 4*A(x))^n = 1 - 3*x - 4*x^3 - 15*x^4 - 76*x^5 - 336*x^6 - 1516*x^7 - 7040*x^8 - 34403*x^9 - 175616*x^10 - 918968*x^11 - 4847040*x^12 + ...
N = Sum_{n>=1} x^(n^2) / (1 - 4*x^n*A(x))^n = x + 4*x^3 + 17*x^4 + 76*x^5 + 336*x^6 + 1516*x^7 + 7040*x^8 + 34401*x^9 + 175616*x^10 + 918968*x^11 + 4847040*x^12 + ...
we see that the sum equals P + N = theta_4(x).
SPECIAL VALUES.
(V.1) A(exp(-Pi)) = 0.05210763699884104351595933706426840151754418802521727110...
where Sum_{n=-oo..+oo} (exp(-n*Pi) - 4*A(exp(-Pi)))^n = (Pi/2)^(1/4)/gamma(3/4) = 0.91357913815611682140724...
(V.2) A(exp(-2*Pi)) = 0.001881490423764068063219673469053308038171175452456126483...
where Sum_{n=-oo..+oo} (exp(-2*n*Pi) - 4*A(exp(-2*Pi)))^n = (Pi/2)^(1/4)/gamma(3/4) * 2^(1/8) = 0.99626511456090713578995...
(V.3) A(exp(-4*Pi)) = 0.000003487391003072013497532566545785034046098962165471423...
where Sum_{n=-oo..+oo} (exp(-4*n*Pi) - 4*A(exp(-4*Pi)))^n = Pi^(1/4)/gamma(3/4) * (sqrt(2) + 1)^(1/4)/2^(7/16) = 0.99999302531528758200931...
(V.4) A(exp(-10*Pi)) = 0.000000000000022711010683243001546817769702787327972263611...
where Sum_{n=-oo..+oo} (exp(-10*n*Pi) - 4*A(exp(-10*Pi)))^n = Pi^(1/4)/gamma(3/4) * 2^(7/8)/((5^(1/4) - 1)*sqrt(5*sqrt(5) + 5)) = 0.99999999999995457797863...

Links

Crossrefs

Cf. A369672 (dual), A002448 (theta_4), A355868.

Programs

  • PARI
    {a(n) = my(A=[1]); for(i=1,n, A = concat(A,0); M=sqrtint(#A+4);
A[#A] = polcoeff( (-sum(n=-M,M, (-1)^n * x^(n^2)) + sum(n=-#A,#A, (x^n - 4*x*Ser(A))^n) )/4, #A); ); A[n]}
for(n=1,30,print1(a(n),", "))
  • PARI
    {a(n) = my(A=[1]); for(i=1,n, A = concat(A,0); M=sqrtint(#A+4);
A[#A] = polcoeff( (-sum(n=-M,M, (-1)^n * x^(n^2)) + sum(n=-#A,#A, x^(n^2)/(1 - 4*x^(n+1)*Ser(A))^n) )/4, #A); ); A[n]}
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} (x^n - 4*A(x))^n = Sum_{n=-oo..+oo} (-1)^n * x^(n^2).
(2) Sum_{n=-oo..+oo} x^n * (x^n + 4*A(x))^(n-1) = Sum_{n=-oo..+oo} (-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} x^(n^2) / (1 - 4*x^n*A(x))^n = Sum_{n=-oo..+oo} (-1)^n * x^(n^2).
(5) Sum_{n=-oo..+oo} x^(n^2) / (1 + 4*x^n*A(x))^(n+1) = Sum_{n=-oo..+oo} (-1)^n * x^(n^2).
(6) Sum_{n=-oo..+oo} (-1)^n * x^(n*(n-1)) / (1 - 4*x^n*A(x))^n = 0.
a(n) ~ c * d^n / n^(3/2), where d = 5.9085050558... and c = 0.2952711268... - Vaclav Kotesovec, Feb 03 2024

A359673 a(n) = coefficient of x^n in A(x) where 1 = Sum_{n=-oo..+oo} (2*x + (-x)^n*A(x)^n)^n.

Original entry on oeis.org

1, 2, 5, 13, 30, 74, 202, 616, 2126, 7828, 29366, 110398, 414214, 1556848, 5892713, 22524354, 86954484, 338421674, 1324660464, 5204326208, 20498580511, 80907096678, 320002290542, 1268500509496, 5040195484362, 20073242195580, 80120884387322, 320442284717582, 1283939790460139
Offset: 0

Views

Author

Paul D. Hanna, Jan 10 2023

Keywords

Comments

Given g.f. A(x), x*A(x) equals a series reversion of x*G(-x) where G(x) is the g.f. of A355868.

Examples

			G.f.: A(x) = 1 + 2*x + 5*x^2 + 13*x^3 + 30*x^4 + 74*x^5 + 202*x^6 + 616*x^7 + 2126*x^8 + 7828*x^9 + 29366*x^10 + 110398*x^11 + 414214*x^12 + ...
SPECIFIC VALUES.
A(x) = 2 at x = 0.2170550872218893465015254812376904599677836767029937...
A(1/5) = 1.8185729641608353079390837085677719656772552871159724...

Links

Crossrefs

Cf. A355868.

Programs

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

Formula

G.f. A(x) satisfies:
(1) 1 = Sum_{n=-oo..+oo} (2*x + (-x)^n * A(x)^n)^n.
(2) 1 = Sum_{n=-oo..+oo} -x^(2*n+1) * A(x)^(n+1) * (2 + (-x)^n * A(x)^(n+1))^n.
(3) 1 = Sum_{n=-oo..+oo} (-x)^(n^2) * A(x)^(n^2) / (1 - 2*(-x)^(n+1) * A(x)^n)^n.
(4) 1 = Sum_{n=-oo..+oo} (-x)^(n^2) * A(x)^(n^2) / (1 + 2*(-x)^(n+1) * A(x)^n)^(n+1).

A369672 Expansion of g.f. A(x) satisfying Sum_{n=-oo..+oo} (-1)^n * (x^n - 4*A(x))^n = theta_3(x).

Original entry on oeis.org

1, -4, 19, -100, 569, -3416, 21302, -136636, 895572, -5971096, 40366463, -276036720, 1905940182, -13269019988, 93040431283, -656472509864, 4657492107245, -33205607204468, 237777067846451, -1709374453370956, 12332468208675821, -89262196983781332, 647988910138661556
Offset: 1

Views

Author

Paul D. Hanna, Feb 03 2024

Keywords

Comments

Note: theta_3(x) = Sum_{n=-oo..+oo} x^(n^2) - see A000122.
Congruences:
(C.1) a(2*n) == 0 (mod 4) for n >= 1.
(C.2) a(n) == A369671(n) (mod 4) for n >= 1.
(C.3) a(2*n)/4 == -A369671(2*n)/4 (mod 4) for n >= 1.

Examples

			G.f.: A(x) = x - 4*x^2 + 19*x^3 - 100*x^4 + 569*x^5 - 3416*x^6 + 21302*x^7 - 136636*x^8 + 895572*x^9 - 5971096*x^10 + 40366463*x^11 - 276036720*x^12 + ...
where Sum_{n=-oo..+oo} (-1)^n * (x^n - 4*A(x))^n = theta_3(x), and
theta_3(x) = 1 + 2*x + 2*x^4 + 2*x^9 + 2*x^16 + 2*x^25 + ... + x^(n^2) + ...
RELATED SERIES.
When we break up the doubly infinite sum into the following parts
P = Sum_{n>=0} (-1)^n * (x^n - 4*A(x))^n = 1 + 3*x + 4*x^3 - 15*x^4 + 92*x^5 - 528*x^6 + 3196*x^7 - 20032*x^8 + 128819*x^9 - 845312*x^10 + 5638568*x^11 - 38122176*x^12 + ...
N = Sum_{n>=1} (-1)^n * x^(n^2) / (1 - 4*x^n*A(x))^n = -x - 4*x^3 + 17*x^4 - 92*x^5 + 528*x^6 - 3196*x^7 + 20032*x^8 - 128817*x^9 + 845312*x^10 - 5638568*x^11 + 38122176*x^12 + ...
we see that the sum equals P + N = theta_3(x).
SPECIAL VALUES.
(V.1) A(exp(-Pi)) = 0.036996905719511834010608252452763733693844226179196126014832...
where Sum_{n=-oo..+oo} (-1)^n * (exp(-n*Pi) - 4*A(exp(-Pi)))^n = Pi^(1/4)/gamma(3/4) = 1.0864348112133080...
(V.2) A(exp(-2*Pi)) = 0.0018536158947374219405603135305712038712234615914707006019...
where Sum_{n=-oo..+oo} (-1)^n * (exp(-2*n*Pi) - 4*A(exp(-2*Pi)))^n = Pi^(1/4)/gamma(3/4) * sqrt(2 + sqrt(2))/2 = 1.0037348854877390...
(V.3) A(exp(-3*Pi)) = 0.0000806734779029429093753810781078431328279003228392603227...
where Sum_{n=-oo..+oo} (-1)^n * (exp(-3*n*Pi) - 4*A(exp(-3*Pi)))^n = Pi^(1/4)/gamma(3/4) * sqrt(1 + sqrt(3))/(108)^(1/8) = 1.000161399035140...
(V.4) A(exp(-4*Pi)) = 0.0000034872937107879617892620501277220047637185282553554945...
where Sum_{n=-oo..+oo} (-1)^n * (exp(-4*n*Pi) - 4*A(exp(-4*Pi)))^n = Pi^(1/4)/gamma(3/4) * (2 + 8^(1/4))/4 = 1.000161399035140...
(V.5) A(exp(-5*Pi)) = 0.0000001507016366950287572418174619564191722052174968450159...
where Sum_{n=-oo..+oo} (-1)^n * (exp(-5*n*Pi) - 4*A(exp(-5*Pi)))^n = Pi^(1/4)/gamma(3/4) * sqrt((2 + sqrt(5))/5) = 1.0000003014034550...

Links

Crossrefs

Cf. A369671 (dual), A000122 (theta_3), A355868.

Programs

  • PARI
    {a(n) = my(A=[1]); for(i=1,n, A = concat(A,0); M=sqrtint(#A+4);
A[#A] = polcoeff( (sum(n=-M,M, x^(n^2)) - sum(n=-#A,#A, (-1)^n * (x^n - 4*x*Ser(A))^n) )/4, #A); ); A[n]}
for(n=1,30,print1(a(n),", "))
  • PARI
    {a(n) = my(A=[1]); for(i=1,n, A = concat(A,0); M=sqrtint(#A+4);
A[#A] = polcoeff( (sum(n=-M,M, x^(n^2)) - sum(n=-#A,#A, (-1)^n * x^(n^2)/(1 - 4*x^(n+1)*Ser(A))^n) )/4, #A); ); A[n]}
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 = Sum_{n=-oo..+oo} x^(n^2).
(2) Sum_{n=-oo..+oo} (-1)^n * x^n * (x^n - 4*A(x))^(n-1) = Sum_{n=-oo..+oo} 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*x^n*A(x))^n = Sum_{n=-oo..+oo} x^(n^2).
(5) Sum_{n=-oo..+oo} (-1)^n * x^(n^2) / (1 - 4*x^n*A(x))^(n+1) = Sum_{n=-oo..+oo} x^(n^2).
(6) Sum_{n=-oo..+oo} (-1)^n * x^(n*(n-1)) / (1 - 4*x^n*A(x))^n = 0.
a(n) ~ c * (-1)^(n+1) * d^n / n^(3/2), where d = 7.7471235933114571108403244715948697607... and c = 0.26329435412874059034137968338302672... - Vaclav Kotesovec, Feb 03 2024

A355867 Coefficients in the even function A(x) = Sum_{n>=0} a(n)*x^(2*n) such that: 2 = Sum_{n=-oo..+oo} x^n * (x^n + i*sqrt(A(x)))^n, where i^2 = -1.

Original entry on oeis.org

1, 1, -1, -6, -3, 27, 64, -72, -580, -573, 3276, 10778, -4429, -94493, -153086, 463061, 2197569, 604351, -17222574, -40338277, 64029441, 477897865, 433963667, -3248816635, -10525409672, 6577294016, 106318417880, 163863253517, -599970596239, -2714863450622
Offset: 0

Views

Author

Paul D. Hanna, Aug 09 2022

Keywords

Comments

What is the pattern to the signs of the terms?
Related identity: Sum_{n=-oo..+oo} (-x)^n * (x^n + y)^n = 0 for all y.
Related identity: Sum_{n=-oo..+oo} (-1)^n * x^(n*(n-1)) / (1 + y*x^n)^n = 0 for all y.

Examples

			G.f.: A(x) = 1 + x^2 - x^4 - 6*x^6 - 3*x^8 + 27*x^10 + 64*x^12 - 72*x^14 - 580*x^16 - 573*x^18 + 3276*x^20 + 10778*x^22 - 4429*x^24 + ...
Let B = sqrt(A(x)) and i = sqrt(-1), then the imaginary part vanishes in the following sums:
(1) 2 = ... + x^(-3)/(x^(-3) + i*B)^3 + x^(-2)/(x^(-2) + i*B)^2 + x^(-1)/(x^(-1) + i*B) + 1 + x*(x + i*B) + x^2*(x^2 + i*B)^2 + x^3*(x^3 + i*B)^3 + ... + x^n*(x^n + i*sqrt(A(x)))^n + ...
(2) 0 = ... - x^(-3)/(x^(-3) + i*B)^3 + x^(-2)/(x^(-2) + i*B)^2 - x^(-1)/(x^(-1) + i*B) + 1 - x*(x + i*B) + x^2*(x^2 + i*B)^2 - x^3*(x^3 + i*B)^3 + ... + (-x)^n*(x^n + i*sqrt(A(x)))^n + ...
(3) 1 = ... + x^(-6)/(x^(-6) + i*B)^6 + x^(-4)/(x^(-4) + i*B)^4 + x^(-2)/(x^(-2) + i*B)^2 + 1 + x^2*(x^2 + i*B)^2 + x^4*(x^4 + i*B)^4 + x^6*(x^6 + i*B)^6 + ... + x^(2*n)*(x^(2*n) + i*sqrt(A(x)))^(2*n) + ...
(4) 1 = ... + x^(-5)/(x^(-5) + i*B)^5 + x^(-3)/(x^(-3) + i*B)^3 + x^(-1)/(x^(-1) + i*B) + x*(x + i*B) + x^3*(x^3 + i*B)^3 + x^5*(x^5 + i*B)^5 + ... + x^(2*n+1)*(x^(2*n+1) + i*sqrt(A(x)))^(2*n+1) + ...
where
B = sqrt(A(x)) = 1 + 2*(x/2)^2 - 10*(x/2)^4 - 172*(x/2)^6 - 90*(x/2)^8 + 12284*(x/2)^10 + 90812*(x/2)^12 - 664088*(x/2)^14 - 14660346*(x/2)^16 - 35699220*(x/2)^18 + 1460864084*(x/2)^20 + ...
The expansion of Sum_{n=-oo..+oo} x^n * (x^n + i*sqrt(A(x)))^n yields
2 = 2 + (2*i^2 + 2)*x^2 + (2*i^4 + 2*i^2)*x^4 + (2*i^6 + 4*i^4 + 4*i^2 + 2)*x^6 + (2*i^8 + 6*i^6 - 2*i^4 - 6*i^2)*x^8 + (2*i^10 + 8*i^8 - 18*i^4 - 12*i^2)*x^10 + (2*i^12 + 10*i^10 + 4*i^8 - 46*i^6 - 14*i^4 + 30*i^2 + 2)*x^12 + (2*i^14 + 12*i^12 + 10*i^10 - 64*i^8 - 76*i^6 + 110*i^4 + 122*i^2)*x^14 + (2*i^16 + 14*i^14 + 18*i^12 - 80*i^10 - 178*i^8 + 210*i^6 + 308*i^4 + 6*i^2)*x^16 + ...
in which all coefficients of x^n evaluate to zero except the constant term.
Specific values.
Let a = A(1/2) = 1.11275889505675972780876...
and b = sqrt(a) = 1.05487387637421362384214...,
then 2 = Sum_{n=-oo..+oo} 1/2^n * (1/2^n + i*b)^n.
The signs of the terms begin:
[+,+,-,-,-,+,+,-,-,-,+,+,-,-,-,+,+,+,-,-,+,+,+,-,-,+,+,+,-,-,-, +,+,-,-,-,+,+,-,-,-,+,+,-,-,-,+,+,+,-,-,+,+,+,-,-,+,+,+,-,-,-, +,+,-,-,-,+,+,-,-,-,+,+,+,-,-,+,+,+,-,-,+,+,+,-,-,-,+,+,-,-,-, +,+,-,-,-,+,+,+,-,-,+,+,+,-,-,+,+,+,-,-,+,+,+,-,-,-,+,+,-,-,-, +,+,-,-,-,+,+,+,-,-,+,+,+,-,-,+,+,+,-,-,-,+,+,-,-,-,+,+,-,-,-, +,+,+,-,-,+,+,+,-,-,+,+,+,-,-,+,+,+,-,-,-,+,+,-,-,-,+,+,-,-,-, +,+,+,-,-,+,+,+,-,-,+,+,+,-,-, ...].

Links

Crossrefs

Cf. A355868.

Programs

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

Formula

G.f. A(x) = Sum_{n>=0} a(n)*x^(2*n) satisfies the following sums.
(1) 2 = Sum_{n=-oo..+oo} x^n * (x^n + i*sqrt(A(x)))^n.
(2) 0 = Sum_{n=-oo..+oo} (-1)^n * x^n * (x^n + i*sqrt(A(x)))^n.
(3) 1 = Sum_{n=-oo..+oo} x^(2*n) * (x^(2*n) + i*sqrt(A(x)))^(2*n).
(4) 1 = Sum_{n=-oo..+oo} x^(2*n+1) * (x^(2*n+1) + i*sqrt(A(x)))^(2*n+1).
(5) 2 = Sum_{n=-oo..+oo} x^(n*(n-1)) / (1 + i*sqrt(A(x))*x^n)^n.
(6) 0 = Sum_{n=-oo..+oo} (-1)^n * x^(n*(n-1)) / (1 + i*sqrt(A(x))*x^n)^n.
(7) 1 = Sum_{n=-oo..+oo} x^(2*n*(2*n-1)) / (1 + i*sqrt(A(x))*x^(2*n))^(2*n).
(8) 1 = Sum_{n=-oo..+oo} x^(2*n*(2*n+1)) / (1 + i*sqrt(A(x))*x^(2*n+1))^(2*n+1).

A359671 a(n) = coefficient of x^n in A(x) where 1 = Sum_{n=-oo..+oo} (x^n - x*A(x))^n.

Original entry on oeis.org

2, 4, 6, 6, 10, 78, 412, 1394, 3312, 6416, 17454, 83334, 384284, 1377888, 3931286, 10234748, 31776266, 127848076, 527518582, 1910397078, 6035143914, 18202417974, 60151348904, 226355566282, 874920531958, 3166323335574, 10599244540550, 34588365630694, 118339356017608
Offset: 0

Views

Author

Paul D. Hanna, Jan 10 2023

Keywords

Comments

All terms are even: a(n) = 2 * A355868(n) for n >= 0.
Related identity: 0 = Sum_{n=-oo..+oo} x^n * (y - x^n)^n, which holds formally for all y.

Examples

			G.f.: A(x) = 2 + 4*x + 6*x^2 + 6*x^3 + 10*x^4 + 78*x^5 + 412*x^6 + 1394*x^7 + 3312*x^8 + 6416*x^9 + 17454*x^10 + 83334*x^11 + 384284*x^12 + ...
SPECIFIC VALUES.
A(x) = 3 at x = 0.1794935271005324391410493657541129782265990045275870...
A(x) = 4 at x = 0.2492900841034309263190875287839455698977108450414094...
A(x) = 5 at x = 0.2676600392887397049709560009239544652896107097280049...

Links

Crossrefs

Cf. A355868.

Programs

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

Formula

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

A370032 Expansion of g.f. A(x) satisfying Sum_{n=-oo..+oo} (x^n + A(x))^n = 1 + 3*Sum_{n>=1} x^(n^2).

Original entry on oeis.org

1, -1, 0, 3, -10, 21, -25, -23, 228, -737, 1479, -1245, -4352, 25206, -72761, 128245, -38615, -697798, 3109043, -8016819, 11763729, 6510069, -108216128, 403917707, -925174519, 1025709534, 2228869018, -16585014721, 53758505915, -107811969706, 69758146717, 478423936550, -2520835801152, 7208714823250
Offset: 1

Views

Author

Paul D. Hanna, Feb 11 2024

Keywords

Examples

			G.f.: A(x) = x - x^2 + 3*x^4 - 10*x^5 + 21*x^6 - 25*x^7 - 23*x^8 + 228*x^9 - 737*x^10 + 1479*x^11 - 1245*x^12 - 4352*x^13 + 25206*x^14 - 72761*x^15 + 128245*x^16 + ...
where
Sum_{n=-oo..+oo} (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 + 3*x^64 + ...

Links

Crossrefs

Cf. A370041, A370030, A370031, A355868, A370033, A370034, A370035, A370036, A370037, A370038, A370039, A370043.

Programs

  • PARI
    {a(n) = my(A=[0, 1]); for(i=1, n, A = concat(A, 0);A[#A] = -polcoeff( sum(m=-#A, #A, (x^m + Ser(A))^m ) - 1 - 3*sum(m=1, #A, 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} (x^n + A(x))^n = 1 + 3*Sum_{n>=1} x^(n^2).
(2) Sum_{n=-oo..+oo} x^n * (x^n - A(x))^(n-1) = 1 + 3*Sum_{n>=1} x^(n^2).
(3) Sum_{n=-oo..+oo} (-1)^n * x^n * (x^n + A(x))^n = 0.
(4) Sum_{n=-oo..+oo} x^(n^2) / (1 + x^n*A(x))^n = 1 + 3*Sum_{n>=1} x^(n^2).
(5) Sum_{n=-oo..+oo} x^(n^2) / (1 - x^n*A(x))^(n+1) = 1 + 3*Sum_{n>=1} x^(n^2).
(6) Sum_{n=-oo..+oo} (-1)^n * x^(n*(n-1)) / (1 + x^n*A(x))^n = 0.

A378829 G.f. A(x) satisfies 1 = Sum_{n=-oo..+oo} (A(x)^n - 2*x)^n.

Original entry on oeis.org

1, -2, 5, -13, 30, -74, 202, -616, 2126, -7828, 29366, -110398, 414214, -1556848, 5892713, -22524354, 86954484, -338421674, 1324660464, -5204326208, 20498580511, -80907096678, 320002290542, -1268500509496, 5040195484362, -20073242195580, 80120884387322, -320442284717582, 1283939790460139
Offset: 1

Views

Author

Paul D. Hanna, Dec 13 2024

Keywords

Comments

A signed version of A359673.

Examples

			G.f.: A(x) = x - 2*x^2 + 5*x^3 - 13*x^4 + 30*x^5 - 74*x^6 + 202*x^7 - 616*x^8 + 2126*x^9 - 7828*x^10 + 29366*x^11 - 110398*x^12 + ...
where 1 = Sum_{n=-oo..+oo} (A(x)^n - 2*x)^n.
SPECIFIC VALUES.
A(t) = 1/6 at t = 0.24134833288352420167420358490093379236139061653959...
  where 1 = Sum_{n=-oo..+oo} (1/6^n - 2*t)^n.
A(t) = 1/7 at t = 0.19473287649699543474178954182484954936895675300220...
  where 1 = Sum_{n=-oo..+oo} (1/7^n - 2*t)^n.
A(t) = 1/8 at t = 0.16330047299490635761734791354706359079698287572429...
  where 1 = Sum_{n=-oo..+oo} (1/8^n - 2*t)^n.
A(t) = exp(-Pi) at t = 0.04720243920412572796492634515550526365563452970121157309...
  where 1 = Sum_{n=-oo..+oo} (exp(-n*Pi) - 2*t)^n,
  also, 1 = Sum_{n=-oo..+oo} exp(-n^2*Pi) / (1 - 2*t*exp(-n*Pi))^n;
  compare to Sum_{n=-oo..+oo} exp(-n^2*Pi) = Pi^(1/4)/gamma(3/4).
A(t) = exp(-2*Pi) at t = 0.001874436990256710694689538031391789940066981740061145959...
  where 1 = Sum_{n=-oo..+oo} (exp(-2*n*Pi) - 2*t)^n,
  also, 1 = Sum_{n=-oo..+oo} exp(-2*n^2*Pi) / (1 - 2*t*exp(-2*n*Pi))^n;
  compare to Sum_{n=-oo..+oo} exp(-2*n^2*Pi) = Pi^(1/4)/gamma(3/4) * sqrt(2+sqrt(2))/2.
A(1/5) = 0.14570268760195709902234365534810153966906514204980...
  where 1 = Sum_{n=-oo..+oo} (A(1/5)^n - 2/5)^n.
A(1/6) = 0.12698642862956730423090954809810167590805619510041...
  where 1 = Sum_{n=-oo..+oo} (A(1/6)^n - 1/3)^n.
A(1/7) = 0.11253270334433369822784652362071431711460474251926...
A(1/8) = 0.10104551587569245791494155789285565556961920656039...
  where 1 = Sum_{n=-oo..+oo} (A(1/8)^n - 1/4)^n.

Links

Crossrefs

Cf. A355868, A359673.

Programs

  • PARI
    {a(n) = my(V=[0,1],A); for(i=1, n, V=concat(V, 0); A=Ser(V);
V[#V] = polcoef( -sum(m=-#V, #V, (A^m - 2*x)^m ), #V-1)/2); V[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) 1 = Sum_{n=-oo..+oo} (A(x)^n - 2*x)^n.
(2) 1 = Sum_{n=-oo..+oo} A(x)^n * (A(x)^n + 2*x)^(n-1).
(3) 0 = Sum_{n=-oo..+oo} (-1)^n * (A(x)^n - 2*x)^(n-1).
(4) 0 = Sum_{n=-oo..+oo} (-1)^n * A(x)^(2*n) * (A(x)^n + 2*x)^(n+1).
(5) 1 = Sum_{n=-oo..+oo} A(x)^(n^2) / (1 - 2*x*A(x)^n)^n.
(6) 1 = Sum_{n=-oo..+oo} A(x)^(n^2) / (1 + 2*x*A(x)^n)^(n+1).
(7) 0 = Sum_{n=-oo..+oo} (-1)^n * A(x)^(n*(n+1)) / (1 + 2*x*A(x)^n)^(n+1).
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