A370038
Expansion of g.f. A(x) satisfying Sum_{n=-oo..+oo} (x^n - 8*A(x))^n = 1 - 6*Sum_{n>=1} x^(n^2).
Original entry on oeis.org
1, 8, 63, 489, 3761, 28911, 224174, 1768801, 14298852, 118834966, 1014912939, 8876489811, 79106007766, 714758437500, 6521121292423, 59905861779190, 553172777516749, 5129986605394544, 47761053650028335, 446350549038171483, 4186889953961917077, 39416115485839527945
Offset: 1
G.f.: A(x) = x + 8*x^2 + 63*x^3 + 489*x^4 + 3761*x^5 + 28911*x^6 + 224174*x^7 + 1768801*x^8 + 14298852*x^9 + 118834966*x^10 + 1014912939*x^11 + ...
where
Sum_{n=-oo..+oo} (x^n - 8*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)) = 0.06579433445460281447496748523290398966344297589844019028...
then Sum_{n=-oo..+oo} (exp(-n*Pi) - 8*A)^n = 4 - 3*Pi^(1/4)/gamma(3/4) = 0.740695566360075956...
(V.2) Let A = A(exp(-2*Pi)) = 0.001895757786183755555448115532175643265455444051246465664...
then Sum_{n=-oo..+oo} (exp(-2*n*Pi) - 8*A)^n = 4 - 3*sqrt(2 + sqrt(2))/2 * Pi^(1/4)/gamma(3/4) = 0.98879534353678272...
(V.3) Let A = A(-exp(-Pi)) = -0.03207876150064786089070312769117792591667175850120792604...
then Sum_{n=-oo..+oo} ((-1)^n*exp(-n*Pi) - 8*A)^n = 4 - 3*(Pi/2)^(1/4)/gamma(3/4) = 1.25926258553164953...
(V.4) Let A = A(-exp(-2*Pi)) = -0.001839948412029108042031275075360099309960919616491079407...
then Sum_{n=-oo..+oo} ((-1)^n*exp(-2*n*Pi) - 8*A)^n = 4 - 3*2^(1/8)*(Pi/2)^(1/4)/gamma(3/4) = 1.01120465631727859...
Cf.
A370041,
A370030,
A370031,
A355868,
A370033,
A370034,
A370035,
A370036,
A370037,
A370039,
A370043.
-
{a(n) = my(A=[0,1]); for(i=1,n, A = concat(A,0);
A[#A] = polcoeff( sum(m=-#A,#A, (x^m - 8*Ser(A))^m ) - 1 + 6*sum(m=1,#A, x^(m^2) ), #A-1)/8 ); A[n+1]}
for(n=1,30, print1(a(n),", "))
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
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...
Cf.
A370041,
A370030,
A370031,
A355868,
A370033,
A370034,
A370035,
A370036,
A370037,
A370038,
A370043.
-
{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),", "))
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
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...
Cf.
A370041,
A370030,
A370031,
A355868,
A370033,
A370034,
A370035,
A370036,
A370037,
A370038,
A370039.
-
{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),", "))
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.
-
/* 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(""))
-
/* 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(""))
A370153
Expansion of g.f. (theta_3(x) - 1)/2 * Product_{n>=1} (1 - x^(4*n-2)) / (1 - x^(4*n)).
Original entry on oeis.org
1, 0, -1, 1, 1, -1, -2, 1, 4, -2, -5, 3, 6, -4, -9, 6, 13, -8, -17, 11, 21, -15, -28, 19, 39, -25, -49, 33, 60, -42, -78, 53, 101, -68, -125, 87, 153, -108, -192, 134, 241, -167, -295, 207, 357, -255, -438, 311, 540, -380, -652, 465, 781, -563, -946, 678, 1145, -819, -1368, 986, 1627
Offset: 1
G.f.: A(x) = x - x^3 + x^4 + x^5 - x^6 - 2*x^7 + x^8 + 4*x^9 - 2*x^10 - 5*x^11 + 3*x^12 + 6*x^13 - 4*x^14 - 9*x^15 + 6*x^16 + 13*x^17 - 8*x^18 + ...
which equals A(x) = P(x) / Q(x)
where
P(x) = x + x^4 + x^9 + x^16 + x^25 + x^36 + x^49 + ...
Q(x) = 1 + x^2 + x^6 + x^12 + x^20 + x^30 + x^42 + ...
-
{a(n) = my(P = sum(m=1,sqrtint(n+1), x^(m^2) +x*O(x^n)),
Q = sum(m=0,sqrtint(n+1), x^(m*(m+1)) +x*O(x^n))); polcoeff(P/Q,n)}
for(n=1,50,print1(a(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
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 + ...
Cf.
A370041,
A370030,
A370031,
A355868,
A370033,
A370034,
A370035,
A370036,
A370037,
A370038,
A370039,
A370043.
-
{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), ", "))
Comments