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
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).
-
{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(""))
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
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...
-
{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),", "))
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
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...
-
{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),", "))
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
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...
-
{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),", "))
A370025
Expansion of g.f. A(x) satisfying Sum_{n=-oo..+oo} (-1)^n * (x^n + 5*A(x))^n = 1 + 7*Sum_{n>=1} (-1)^n * x^(n^2).
Original entry on oeis.org
1, 5, 28, 169, 1054, 6667, 42627, 275211, 1791132, 11731613, 77242391, 510826889, 3391115560, 22586150402, 150866419771, 1010290295683, 6780795305121, 45602955247738, 307252705965207, 2073546683753911, 14014659243408481, 94851805738129599, 642767262413178788, 4360774590348465669
Offset: 1
G.f.: A(x) = x + 5*x^2 + 28*x^3 + 169*x^4 + 1054*x^5 + 6667*x^6 + 42627*x^7 + 275211*x^8 + 1791132*x^9 + 11731613*x^10 + 77242391*x^11 + 510826889*x^12 + ...
where
Sum_{n=-oo..+oo} (-1)^n * (x^n + 5*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)) = 0.05561899448885311185126383683351896617798185829954077412...
then Sum_{n=-oo..+oo} (-1)^n * (exp(-n*Pi) + 5*A)^n = (7*(Pi/2)^(1/4)/gamma(3/4) - 5)/2 = 0.69752698354640887492534...
(V.2) Let A = A(exp(-2*Pi)) = 0.001885063870555508038278982205994616246272805466135524875...
then Sum_{n=-oo..+oo} (-1)^n * (exp(-2*n*Pi) + 5*A)^n = (7*2^(1/8)*(Pi/2)^(1/4)/gamma(3/4) - 5)/2 = 0.986927900963174975264...
(V.3) Let A = A(-exp(-Pi)) = -0.03567173485605183837843763169616623725553901880108539739...
then Sum_{n=-oo..+oo} (-1)^n * ((-1)^n*exp(-n*Pi) + 5*A)^n = (7*Pi^(1/4)/gamma(3/4) - 5)/2 = 1.302521839246578051013...
(V.4) Let A = A(-exp(-2*Pi)) = -0.0018536159060139689998658447922411419770684746756327600438...
then Sum_{n=-oo..+oo} (-1)^n * ((-1)^n*exp(-2*n*Pi) + 5*A)^n = (7*sqrt(2 + sqrt(2))/2 * Pi^(1/4)/gamma(3/4) - 5)/2 = 1.01307209920708681866...
-
{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 + 5*Ser(A))^m ) - 1 - 7*sum(m=1,#A, (-1)^m * x^(m^2) ), #A-1)/5 ); A[n+1]}
for(n=1,30, print1(a(n),", "))
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
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...
-
{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),", "))
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
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...
-
{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),", "))
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
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...
-
{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),", "))
A370040
Triangle of coefficients T(n,k) in g.f. A(x,y) satisfying Sum_{n=-oo..+oo} (-1)^n * (x^n + y*A(x,y))^n = 1 + (y+2)*Sum_{n>=1} (-1)^n * x^(n^2), for n >= 1, as read by rows.
Original entry on oeis.org
1, 0, 1, 3, 0, 1, -1, 9, 0, 1, 9, -6, 18, 0, 1, -3, 54, -19, 30, 0, 1, 22, -54, 185, -44, 45, 0, 1, -9, 264, -294, 475, -85, 63, 0, 1, 52, -324, 1463, -1026, 1020, -146, 84, 0, 1, -22, 1127, -2715, 5531, -2781, 1939, -231, 108, 0, 1, 111, -1534, 9648, -13430, 16470, -6384, 3374, -344, 135, 0, 1, -51, 4338, -19005, 51853, -49032, 41567, -13020, 5490, -489, 165, 0, 1
Offset: 1
G.f.: A(x,y) = x*(1) + x^2*(0 + y) + x^3*(3 + y^2) + x^4*(-1 + 9*y + y^3) + x^5*(9 - 6*y + 18*y^2 + y^4) + x^6*(-3 + 54*y - 19*y^2 + 30*y^3 + y^5) + x^7*(22 - 54*y + 185*y^2 - 44*y^3 + 45*y^4 + y^6) + x^8*(-9 + 264*y - 294*y^2 + 475*y^3 - 85*y^4 + 63*y^5 + y^7) + x^9*(52 - 324*y + 1463*y^2 - 1026*y^3 + 1020*y^4 - 146*y^5 + 84*y^6 + y^8) + x^10*(-22 + 1127*y - 2715*y^2 + 5531*y^3 - 2781*y^4 + 1939*y^5 - 231*y^6 + 108*y^7 + y^9) + ...
where
Sum_{n=-oo..+oo} (-1)^n * (x^n + y*A(x,y))^n = 1 + (y+2)*Sum_{n>=1} (-1)^n * x^(n^2).
TRIANGLE.
This triangle of coefficients T(n,k) of x^n*y^k in g.f. A(x,y) begins
1;
0, 1;
3, 0, 1;
-1, 9, 0, 1;
9, -6, 18, 0, 1;
-3, 54, -19, 30, 0, 1;
22, -54, 185, -44, 45, 0, 1;
-9, 264, -294, 475, -85, 63, 0, 1;
52, -324, 1463, -1026, 1020, -146, 84, 0, 1;
-22, 1127, -2715, 5531, -2781, 1939, -231, 108, 0, 1;
111, -1534, 9648, -13430, 16470, -6384, 3374, -344, 135, 0, 1;
-51, 4338, -19005, 51853, -49032, 41567, -13020, 5490, -489, 165, 0, 1;
230, -6274, 55413, -128974, 208178, -146098, 92869, -24300, 8475, -670, 198, 0, 1; ...
Cf.
A370020,
A370021,
A370022,
A370023,
A370024,
A370025,
A370026,
A370027,
A370028,
A370029,
A370042.
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/* Generate A(x,y) by use of definition in name */
{T(n,k) = my(A=[0,1]); for(i=1,n, A = concat(A,0);
A[#A] = polcoeff( sum(m=-#A,#A, (-1)^m * (x^m + y*Ser(A))^m ) - 1 - (y+2)*sum(m=1,#A, (-1)^m * x^(m^2) ), #A-1)/y ); H=A; polcoeff(A[n+1],k,y)}
for(n=1,15, for(k=0,n-1, print1(T(n,k),", "));print(""))
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/* Generate A(x,y) recursively using integration wrt y */
{T(n,k) = my(A = x +x*O(x^n), M=sqrtint(n+1), Q = sum(m=1,M, (-1)^m * x^(m^2)) +x*O(x^n));
for(i=0,n, A = (1/y) * intformal( Q / sum(m=-M,n, (-1)^m * m * (x^m + y*A)^(m-1)), y) +x*O(x^n));
polcoeff(polcoeff(A,n,x),k,y)}
for(n=1,15, for(k=0,n-1, print1(T(n,k),", "));print(""))
Showing 1-9 of 9 results.
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