A384271
G.f. A(x) satisfies -x = Product_{n>=1} (1 - x^n/A(x)) * (1 - x^(n-1)*A(x)) * (1 + x^n).
Original entry on oeis.org
1, 1, 1, 3, 5, 14, 31, 85, 214, 589, 1572, 4385, 12124, 34315, 97006, 277958, 797969, 2310313, 6708311, 19590928, 57386238, 168805975, 497956135, 1473704926, 4372436946, 13007158125, 38779605810, 115872525324, 346897113802, 1040486309806, 3126167631775, 9407946523434, 28355033124335, 85582565615778
Offset: 0
G.f.: A(x) = 1 + x + x^2 + 3*x^3 + 5*x^4 + 14*x^5 + 31*x^6 + 85*x^7 + 214*x^8 + 589*x^9 + 1572*x^10 + 4385*x^11 + 12124*x^12 + ...
RELATED SERIES.
1/A(x) = 1 - x - 2*x^3 - 7*x^5 - 4*x^6 - 33*x^7 - 43*x^8 - 190*x^9 - 363*x^10 - 1265*x^11 - 2967*x^12 - 9313*x^13 - 24254*x^14 + ...
By definition of g.f. A(x),
-x = (1 - x/A(x))*(1 - A(x))*(1 + x) * (1 - x^2/A(x))*(1 - x*A(x))*(1 + x^2) * (1 - x^3/A(x))*(1 - x^2*A(x))*(1 + x^3) * (1 - x^4/A(x))*(1 - x^3*A(x))*(1 + x^4) * (1 - x^5/A(x))*(1 - x^4*A(x))*(1 + x^5) * (1 - x^6/A(x))*(1 - x^5*A(x))*(1 + x^6) * ...
also,
-x*theta_4(x) = (1 - x/A(x))*(1 - A(x))*(1 - x) * (1 - x^2/A(x))*(1 - x*A(x))*(1 - x^2) * (1 - x^3/A(x))*(1 - x^2*A(x))*(1 - x^3) * (1 - x^4/A(x))*(1 - x^3*A(x))*(1 - x^4) * (1 - x^5/A(x))*(1 - x^4*A(x))*(1 - x^5) * (1 - x^6/A(x))*(1 - x^5*A(x))*(1 - x^6) * ...
where Jacobi's theta_4(x) begins
theta_4(x) = 1 - 2*x + 2*x^4 - 2*x^9 + 2*x^16 - 2*x^25 + 2*x^36 - 2*x^49 +- ... + (-1)^n*2*x^(n^2) + ...
SPECIFIC VALUES.
A(exp(-Pi)) = 1.0453432348429282081117266580603161092013621219944501002...
where Sum_{n=-oo..+oo} (-1)^n * exp(-Pi*n*(n-1)/2) * A(exp(-Pi))^n = -exp(-Pi) * (Pi/2)^(1/4) / gamma(3/4) = -0.03947933420376592813...
A(-exp(-Pi)) = 0.958426933091195985748561440955710208995111661258536170...
where Sum_{n=-oo..+oo} (-1)^(n*(n+1)/2) * exp(-Pi*n*(n-1)/2) * A(-exp(-Pi))^n = exp(-Pi) * Pi^(1/4) / gamma(3/4) = 0.04694910513068872743...
A(t) = 2 at t = 0.31637346425553975249950084871655397381494910538235011...
A(t) = 7/4 at t = 0.306394408393287726599555143524924576884132332626742...
A(t) = 5/3 at t = 0.298403642258683011765026172638519982558475148161098...
A(t) = 3/2 at t = 0.271351341798078045586394278854619398226629821704419...
A(t) = 4/3 at t = 0.222121640630627872529588897705597278294416500502588...
A(t) = 5/4 at t = 0.185212111226798258067304643213927542314746099159395...
A(1/4) = 1.415936196810577322060687637240440296052642753467849...
A(1/5) = 1.280767471524969264389815996502959550941291484191129...
A(1/6) = 1.215194363106761985540779108431983083763494550900814...
A(1/7) = 1.175354795171732738951963612236910785381681269370988...
A(1/8) = 1.148310502549415307985734864677154956069415167149368...
-
(* Calculation of constants {d, c}: *) {1/r, -s*Log[r] * Sqrt[((s-1)*(-2*r*(s-1) * QPochhammer[s, r] * Derivative[0, 1][QPochhammer][-1, r] + s*QPochhammer[-1, r]^2*QPochhammer[s, r]^2 * Derivative[0, 1][QPochhammer][1/s, r] + 2*(s-1) * QPochhammer[-1, r] * (QPochhammer[s, r] - r*Derivative[0, 1][QPochhammer][s, r]))) / (QPochhammer[-1, r] * QPochhammer[s, r] * (-s*Log[r]^2 + (s-1)^2 * QPolyGamma[1, -Log[s]/Log[r], r] + (s-1)^2 * QPolyGamma[1, Log[s]/Log[r], r]))] / (2*Sqrt[Pi])} /. FindRoot[{s * QPochhammer[-1, r] * QPochhammer[1/s, r] * QPochhammer[s, r] == -2*r*(s-1), Log[r]/(s-1) == QPolyGamma[0, Log[1/s]/Log[r], r] - QPolyGamma[0, Log[s]/Log[r], r]}, {r, 1/3}, {s, 2}, WorkingPrecision -> 120] (* Vaclav Kotesovec, May 25 2025 *)
-
{a(n) = my(A=[1,1]); for(i=2,n, A=concat(A,0);
A[#A] = polcoef(x + prod(n=1,#A, (1 - x^n/Ser(A)) * (1 - x^(n-1)*Ser(A)) * (1 + x^n) ),#A-1); ); A[n+1]}
for(n=0,30,print1(a(n),", "))
A384272
G.f. A(x) satisfies -2*x = Product_{n>=1} (1 - x^n/A(x)) * (1 - x^(n-1)*A(x)) * (1 + x^n).
Original entry on oeis.org
1, 2, 2, 6, 16, 50, 144, 478, 1510, 5116, 17034, 58812, 202166, 709228, 2489546, 8848146, 31525526, 113236920, 407983964, 1478249454, 5372468156, 19607233026, 71758722172, 263480958508, 969856453650, 3579426292768, 13239549874552, 49078409375334, 182282423994240, 678289439131812, 2528257204808848
Offset: 0
G.f.: A(x) = 1 + 2*x + 2*x^2 + 6*x^3 + 16*x^4 + 50*x^5 + 144*x^6 + 478*x^7 + 1510*x^8 + 5116*x^9 + 17034*x^10 + 58812*x^11 + 202166*x^12 + ...
where
-2*x = (1 - x/A(x))*(1 - A(x))*(1+x) * (1 - x^2/A(x))*(1 - x*A(x))*(1+x^2) * (1 - x^3/A(x))*(1 - x^2*A(x))*(1+x^3) * (1 - x^4/A(x))*(1 - x^3*A(x))*(1+x^4) * (1 - x^5/A(x))*(1 - x^4*A(x))*(1+x^5) * ...
-
(* Calculation of constants {d,c}: *) With[{k = 2}, Chop[{1/r, (1/Sqrt[2*Pi])*(-1 + s)* Sqrt[(s^2*(-r + s)*Log[r]*((r - s)*Log[1 - r] - r*Log[r] + (r - s)*(QPolyGamma[0, -1 + Log[s]/Log[r], r] + r*Log[r]*(Derivative[0, 1][QPochhammer][-1, r]/ QPochhammer[-1, r] + Derivative[0, 1][QPochhammer][1/s, r]/ QPochhammer[1/s, r] + Derivative[0, 1][QPochhammer][s/r, r]/ QPochhammer[s/r, r])))) / (-2*s*(r + r^2 - 3*r*s + s^3)* Log[r]^2 + 2*(-1 + s)*(-r + s)*(-r + s^2)* Log[r]*(QPolyGamma[0, -Log[s]/Log[r], r] - QPolyGamma[0, -1 + Log[s]/Log[r], r]) + (r - s)^2*(-1 + s)^2*((QPolyGamma[0, -Log[s]/Log[r], r] - QPolyGamma[0, -1 + Log[s]/Log[r], r]) * (Log[r] - QPolyGamma[0, -Log[s]/Log[r], r] + QPolyGamma[0, -1 + Log[s]/Log[r], r]) + QPolyGamma[1, -Log[s]/Log[r], r] + QPolyGamma[1, -1 + Log[s]/Log[r], r]))]} /. FindRoot[{2*k* r + (r*s*QPochhammer[-1, r]*QPochhammer[1/s, r]* QPochhammer[s/r, r])/((r - s)*(-1 + s)) == 0, (-r + s^2)*Log[r] + (r - s)*(-1 + s) * QPolyGamma[0, Log[1/s]/Log[r], r] - (r - s)*(-1 + s)*QPolyGamma[0, Log[s/r]/Log[r], r] == 0}, {r, 1/4}, {s, 2}, WorkingPrecision -> 120]]] (* Vaclav Kotesovec, Jun 30 2025 *)
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{a(n) = my(A=[1,2]); for(i=2,n, A=concat(A,0);
A[#A] = polcoef(2*x + prod(n=1,#A, (1 - x^n/Ser(A)) * (1 - x^(n-1)*Ser(A)) * (1 + x^n) ),#A-1); ); A[n+1]}
for(n=0,30,print1(a(n),", "))
A384273
G.f. A(x) satisfies -3*x = Product_{n>=1} (1 - x^n/A(x)) * (1 - x^(n-1)*A(x)) * (1 + x^n).
Original entry on oeis.org
1, 3, 3, 9, 39, 108, 387, 1581, 5196, 21573, 82596, 318279, 1303146, 5182389, 20919156, 86577264, 351929133, 1462075095, 6077250693, 25277372124, 106131459906, 445859648019, 1878449392365, 7955646845046, 33707865532680, 143344958486019, 610977896794104, 2608218534504888, 11162376089875158
Offset: 0
G.f.: A(x) = 1 + 3*x + 3*x^2 + 9*x^3 + 39*x^4 + 108*x^5 + 387*x^6 + 1581*x^7 + 5196*x^8 + 21573*x^9 + 82596*x^10 + 318279*x^11 + 1303146*x^12 + ...
where
-3*x = (1 - x/A(x))*(1 - A(x))*(1+x) * (1 - x^2/A(x))*(1 - x*A(x))*(1+x^2) * (1 - x^3/A(x))*(1 - x^2*A(x))*(1+x^3) * (1 - x^4/A(x))*(1 - x^3*A(x))*(1+x^4) * (1 - x^5/A(x))*(1 - x^4*A(x))*(1+x^5) * ...
-
(* Calculation of constants {d,c}: *) With[{k = 3}, Chop[{1/r, (1/Sqrt[2*Pi])*(-1 + s)* Sqrt[(s^2*(-r + s)*Log[r]*((r - s)*Log[1 - r] - r*Log[r] + (r - s)*(QPolyGamma[0, -1 + Log[s]/Log[r], r] + r*Log[r]*(Derivative[0, 1][QPochhammer][-1, r]/ QPochhammer[-1, r] + Derivative[0, 1][QPochhammer][1/s, r]/ QPochhammer[1/s, r] + Derivative[0, 1][QPochhammer][s/r, r]/ QPochhammer[s/r, r])))) / (-2*s*(r + r^2 - 3*r*s + s^3) * Log[r]^2 + 2*(-1 + s)*(-r + s)*(-r + s^2)* Log[r]*(QPolyGamma[0, -Log[s]/Log[r], r] - QPolyGamma[0, -1 + Log[s]/Log[r], r]) + (r - s)^2*(-1 + s)^2*((QPolyGamma[0, -Log[s]/Log[r], r] - QPolyGamma[0, -1 + Log[s]/Log[r], r]) * (Log[r] - QPolyGamma[0, -Log[s]/Log[r], r] + QPolyGamma[0, -1 + Log[s]/Log[r], r]) + QPolyGamma[1, -Log[s]/Log[r], r] + QPolyGamma[1, -1 + Log[s]/Log[r], r]))]} /. FindRoot[{2*k* r + (r*s*QPochhammer[-1, r]*QPochhammer[1/s, r] * QPochhammer[s/r, r])/((r - s)*(-1 + s)) == 0, (-r + s^2)*Log[r] + (r - s)*(-1 + s) * QPolyGamma[0, Log[1/s]/Log[r], r] - (r - s)*(-1 + s)*QPolyGamma[0, Log[s/r]/Log[r], r] == 0}, {r, 1/4}, {s, 2}, WorkingPrecision -> 120]]] (* Vaclav Kotesovec, Jun 30 2025 *)
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{a(n) = my(A=[1,3]); for(i=2,n, A=concat(A,0);
A[#A] = polcoef(3*x + prod(n=1,#A, (1 - x^n/Ser(A)) * (1 - x^(n-1)*Ser(A)) * (1 + x^n) ),#A-1); ); A[n+1]}
for(n=0,30,print1(a(n),", "))
A356508
G.f. A(x) satisfies: 2 = Product_{n>=1} (1 + x^n*A(x)) * (1 + x^(n-1)/A(x)).
Original entry on oeis.org
1, 4, 14, 84, 444, 2928, 18214, 125428, 844534, 5989816, 42186878, 305757288, 2215509018, 16326672796, 120612763510, 900561207232, 6746557569136, 50906726784700, 385432963013140, 2933390906035044, 22395805754363208, 171660252748284852, 1319474586701337644
Offset: 0
G.f.: A(x) = 1 + 4*x + 14*x^2 + 84*x^3 + 444*x^4 + 2928*x^5 + 18214*x^6 + 125428*x^7 + 844534*x^8 + 5989816*x^9 + 42186878*x^10 + ...
such that
2 = (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)) * (1 + x^5*A(x))*(1 + x^4/A(x)) * ...
also,
2/P(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 + ... + x^(n*(n+1)/2) * A(x)^n + ...
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 + ...
and
2/P(x) = 2 - 2*x - 2*x^2 + 2*x^5 + 2*x^7 - 2*x^12 - 2*x^15 + 2*x^22 + 2*x^26 - 2*x^35 - 2*x^40 + 2*x^51 + 2*x^57 - 2*x^70 - 2*x^77 + 2*x^92 + 2*x^100 + ...
-
(* Calculation of constants {d,c}: *) {1/r, -s*Log[r]/2 * Sqrt[-(r*(1 + s)*(QPochhammer[-s, r]^2* Derivative[0, 1][QPochhammer][-1/s, r] + 2*(1 + s) * Derivative[0, 1][QPochhammer][-s, r])) / (Pi * QPochhammer[-s, r] * (s* Log[r]^2 + (1 + s)^2*(QPolyGamma[1, Log[-1/s]/Log[r], r] + QPolyGamma[1, Log[-s]/Log[r], r])))]} /. FindRoot[{QPochhammer[-1/s, r]*QPochhammer[-s, r] == 2*(1 + s), (1 + s)*(QPolyGamma[0, Log[-1/s]/Log[r], r] - QPolyGamma[0, Log[-s]/Log[r], r]) == s*Log[r]}, {r, 1/8}, {s, 3}, WorkingPrecision -> 120] (* Vaclav Kotesovec, Sep 30 2023 *)
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{a(n) = my(A=[1]); for(i=1, n, A=concat(A, 0);
A[#A] = polcoeff( -2 + prod(n=1, #A, (1 + x^n*Ser(A)) * (1 + x^(n-1)/Ser(A)) ), #A-1)); A[n+1]}
for(n=0, 30, print1(a(n), ", "))
-
{a(n) = my(A=[1], M, P=prod(k=1, n, 1-x^k +x*O(x^n))); for(i=1, n, A=concat(A, 0); M = ceil(sqrt(2*n+9));
A[#A] = polcoeff( -2*P + sum(m=-M, M, x^(m*(m+1)/2)*Ser(A)^m ), #A-1)); A[n+1]}
for(n=0, 30, print1(a(n), ", "))
A384269
G.f. A(x) satisfies x = Product_{n>=1} (1 - x^n*A(x)) * (1 - x^(n-1)/A(x)) * (1 + x^n).
Original entry on oeis.org
1, 1, 2, 6, 16, 49, 154, 513, 1747, 6078, 21439, 76607, 276685, 1008781, 3707512, 13721086, 51088860, 191245836, 719333008, 2717229481, 10303797518, 39208957744, 149676496756, 573037914270, 2199735075908, 8464921506665, 32648239747059, 126185248269567, 488657718553676, 1895790377527674
Offset: 0
G.f.: A(x) = 1 + x + 2*x^2 + 6*x^3 + 16*x^4 + 49*x^5 + 154*x^6 + 513*x^7 + 1747*x^8 + 6078*x^9 + 21439*x^10 + ...
RELATED SERIES.
1/A(x) = 1 - x - x^2 - 3*x^3 - 5*x^4 - 16*x^5 - 45*x^6 - 155*x^7 - 512*x^8 - 1763*x^9 - 6084*x^10 + ...
SPECIFIC VALUES.
A(exp(-Pi)) = 1.0474973549949421045732567080496722542518531011526934631...
where Sum_{n=-oo..+oo} (-1)^n * exp(-Pi*n*(n+1)/2) * A(exp(-Pi))^n = exp(-Pi) * (Pi/2)^(1/4) / gamma(3/4) = 0.03947933420376592813...
A(-exp(-Pi)) = 0.960086060200580366759936974556134222228793624085744940...
where Sum_{n=-oo..+oo} (-1)^(n*(n-1)/2) * exp(-Pi*n*(n+1)/2) * A(-exp(-Pi))^n = -exp(-Pi) * Pi^(1/4) / gamma(3/4) = -0.04694910513068872743...
A(t) = 2 at t = 0.24484187571695922418922496399796775078115821427621282...
A(t) = 7/4 at t = 0.239324355731620083092236573970947000576283799760943...
A(t) = 5/3 at t = 0.234439889083627870257298020352799276294012688627782...
A(t) = 3/2 at t = 0.217134571709901433113197085617818478214816713922905...
A(t) = 4/3 at t = 0.183806911401666173138177455971709388630788740531594...
A(t) = 5/4 at t = 0.157416870441717618165825450612923233287765184975643...
A(1/5) = 1.401449039483961854381757985869052435618161722574956...
A(1/6) = 1.276318946972284528693666572724710434062725174240448...
A(1/7) = 1.213287805382388838362413216213677242108560133326140...
A(1/8) = 1.174388177498186580244775740286834758637341200438483...
A(1/9) = 1.147764942051942680447238410304951699474657455354304...
-
(* Calculation of constants {d,c}: *) {1/r, -s*Log[r]/2* Sqrt[((s-1)*(-2*r*(s - 1) * QPochhammer[s, r] * Derivative[0, 1][QPochhammer][-1, r] + QPochhammer[-1, r]^2 * QPochhammer[s, r]^2 * Derivative[0, 1][QPochhammer][1/s, r] + 2*(s-1)* QPochhammer[-1, r] * (QPochhammer[s, r] - r*Derivative[0, 1][QPochhammer][s, r])))/ (Pi*QPochhammer[-1, r] * QPochhammer[s, r] * (-s*Log[ r]^2 + (s-1)^2 * (QPolyGamma[1, -Log[s]/Log[r], r] + QPolyGamma[1, Log[s]/Log[r], r])))]} /. FindRoot[{QPochhammer[-1, r] * QPochhammer[1/s, r] * QPochhammer[s, r] == 2*r*(1 - s), s*Log[r] + (s-1) * (QPolyGamma[0, Log[s]/Log[r], r] - QPolyGamma[0, Log[1/s]/Log[r], r]) == 0}, {r, 1/4}, {s, 2}, WorkingPrecision -> 120] (* Vaclav Kotesovec, Jun 01 2025 *)
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{a(n) = my(A=[1,1]); for(i=2,n, A=concat(A,0);
A[#A] = polcoef(x - prod(n=1,#A, (1 - x^n*Ser(A)) * (1 - x^(n-1)/Ser(A)) * (1 + x^n) ),#A-1); ); H=A; A[n+1]}
for(n=0,30,print1(a(n),", "))
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