A384821
G.f. A(x) satisfies -1/x = Sum_{n=-oo..+oo} A(x)^n * x^n * (1 - x^n)^(n+2).
Original entry on oeis.org
1, 2, 5, 22, 91, 416, 1978, 9738, 49181, 253572, 1328528, 7053672, 37866294, 205188765, 1120824743, 6165155890, 34119043994, 189839648588, 1061344406923, 5959197795092, 33588952625106, 189986944364176, 1078034452020854, 6134848540680166, 35005230073846833, 200229444332667654
Offset: 0
G.f.: A(x) = 1 + 2*x + 5*x^2 + 22*x^3 + 91*x^4 + 416*x^5 + 1978*x^6 + 9738*x^7 + 49181*x^8 + 253572*x^9 + 1328528*x^10 + ...
SPECIFIC VALUES.
A(t) = 2 at t = 0.162924020448782314256916956456618618555937137963260...
A(t) = 9/5 at t = 0.15713093477961462528780113190237390843002535981643...
A(t) = 8/5 at t = 0.14467881602482935797425598908263109752382579929421...
A(t) = 3/2 at t = 0.13461615563760120581581313629107981605312435881819...
A(t) = 4/3 at t = 0.10915621052082212882653574706851509193398803739915...
A(1/7) = 1.5793911503434252677981671019480264164820055324466...
A(1/8) = 1.4268350851974567615394958810072981944850896947894...
A(1/9) = 1.3435470274993477728207146854713823085043981519155...
A(1/10) = 1.2892440747830023480637465318368592024118039394009...
A(1/11) = 1.2505209808081799972669805855553805055082827658365...
-
{a(n) = my(A=[1,2,0]); for(i=1, n, A = concat(A, 0);
A[#A-1] = polcoeff( sum(m=-#A, #A, x^m * Ser(A)^m * (1 - x^m +x*O(x^n))^(m+2) ), #A-4)); A[n+1]}
for(n=0, 30, print1(a(n), ", "))
A384822
G.f. A(x) satisfies 1/x^5 = Sum_{n=-oo..+oo} A(x)^n * x^n * (1 - x^n)^(n+4).
Original entry on oeis.org
1, 1, 5, 19, 109, 598, 3592, 22110, 140467, 911136, 6014277, 40260501, 272682397, 1865181921, 12866239311, 89403333632, 625211046931, 4396844409898, 31075863324446, 220618909826500, 1572549447431889, 11249693613964519, 80743512234554655, 581272589032594530, 4196118995069449989
Offset: 0
G.f.: A(x) = 1 + x + 5*x^2 + 19*x^3 + 109*x^4 + 598*x^5 + 3592*x^6 + 22110*x^7 + 140467*x^8 + 911136*x^9 + 6014277*x^10 + ...
-
{a(n) = my(A=[1,1,0,0]); for(i=1, n, A = concat(A, 0);
A[#A-2] = -polcoeff( sum(m=-#A, #A, x^m * Ser(A)^m * (1 - x^m +x*O(x^n))^(m+4) ), #A-9)); A[n+1]}
for(n=0, 30, print1(a(n), ", "))
A384823
G.f. A(x) satisfies -1/x^11 = Sum_{n=-oo..+oo} A(x)^n * x^n * (1 - x^n)^(n+6).
Original entry on oeis.org
1, 1, 4, 28, 173, 1262, 9593, 75928, 618342, 5149640, 43650123, 375347585, 3266282211, 28709930633, 254526671024, 2273271614848, 20435110855838, 184745786960642, 1678668998195885, 15321962225034079, 140418372363945954, 1291587696225346583, 11919771215919819476, 110338977972166474055
Offset: 0
G.f.: A(x) = 1 + x + 4*x^2 + 28*x^3 + 173*x^4 + 1262*x^5 + 9593*x^6 + 75928*x^7 + 618342*x^8 + 5149640*x^9 + 43650123*x^10 + ...
-
{a(n) = my(A=[1,1,0,0,0]); for(i=1, n, A = concat(A, 0);
A[#A-3] = polcoeff( sum(m=-#A, #A, x^m * Ser(A)^m * (1 - x^m +x*O(x^n))^(m+6) ), #A-16)); A[n+1]}
for(n=0, 30, print1(a(n), ", "))
A384824
G.f. A(x) satisfies 1/x^19 = Sum_{n=-oo..+oo} A(x)^n * x^n * (1 - x^n)^(n+8).
Original entry on oeis.org
1, 1, 5, 38, 319, 2871, 27507, 273925, 2808973, 29457644, 314470771, 3405995019, 37334767867, 413397265017, 4617060957512, 51951448775027, 588371324004508, 6701761863368579, 76723673176823126, 882342098781937683, 10188542630975395255, 118082022786322630334, 1373108879790849494070
Offset: 0
G.f.: A(x) = 1 + x + 5*x^2 + 38*x^3 + 319*x^4 + 2871*x^5 + 27507*x^6 + 273925*x^7 + 2808973*x^8 + 29457644*x^9 + 314470771*x^10 + ...
-
{a(n) = my(A=[1,1,0,0,0,0]); for(i=1, n, A = concat(A, 0);
A[#A-4] = -polcoeff( sum(m=-#A, #A, x^m * Ser(A)^m * (1 - x^m +x*O(x^n))^(m+8) ), #A-25)); A[n+1]}
for(n=0, 30, print1(a(n), ", "))
A384825
G.f. A(x) satisfies -1/x^29 = Sum_{n=-oo..+oo} A(x)^n * x^n * (1 - x^n)^(n+10).
Original entry on oeis.org
1, 1, 6, 54, 542, 5955, 69114, 835140, 10391843, 132262619, 1713785727, 22531557603, 299817809184, 4030217936308, 54646151953660, 746513545616000, 10264746883787021, 141955200254335604, 1973170863256461516, 27551902179444882489, 386288077655575999571, 5435910477286670671340
Offset: 0
G.f.: A(x) = 1 + x + 6*x^2 + 54*x^3 + 542*x^4 + 5955*x^5 + 69114*x^6 + 835140*x^7 + 10391843*x^8 + 132262619*x^9 + 1713785727*x^10 + ...
-
{a(n) = my(A=[1,1,0,0,0,0,0]); for(i=1, n, A = concat(A, 0);
A[#A-5] = polcoeff( sum(m=-#A, #A, x^m * Ser(A)^m * (1 - x^m +x*O(x^n))^(m+10) ), #A-36)); A[n+1]}
for(n=0, 30, print1(a(n), ", "))
A384826
G.f. A(x) satisfies 1/x^41 = Sum_{n=-oo..+oo} A(x)^n * x^n * (1 - x^n)^(n+12).
Original entry on oeis.org
1, 1, 7, 73, 861, 11112, 151828, 2159179, 31627690, 473917665, 7230164079, 111926802631, 1753762735460, 27760507986844, 443257137593369, 7130838718144623, 115469073853104486, 1880570694656739472, 30784302913287253256, 506228988080918570208, 8358750672258509735440, 138528877561300962357350
Offset: 0
G.f.: A(x) = 1 + x + 7*x^2 + 73*x^3 + 861*x^4 + 11112*x^5 + 151828*x^6 + 2159179*x^7 + 31627690*x^8 + 473917665*x^9 + 7230164079*x^10 + ...
-
{a(n) = my(A=[1,1,0,0,0,0,0,0]); for(i=1, n, A = concat(A, 0);
A[#A-6] = -polcoeff( sum(m=-#A, #A, x^m * Ser(A)^m * (1 - x^m +x*O(x^n))^(m+12) ), #A-49)); A[n+1]}
for(n=0, 30, print1(a(n), ", "))
A384827
G.f. A(x) satisfies -1/x^55 = Sum_{n=-oo..+oo} A(x)^n * x^n * (1 - x^n)^(n+14).
Original entry on oeis.org
1, 1, 8, 95, 1288, 19116, 300511, 4918268, 82918049, 1430142380, 25115651237, 447578072658, 8073426806649, 147122009148252, 2704441907759235, 50088849266618466, 933792151007378231, 17509062834076661230, 329985690688947517626, 6247533413700369107192, 118768564127167799819733
Offset: 0
G.f.: A(x) = 1 + x + 8*x^2 + 95*x^3 + 1288*x^4 + 19116*x^5 + 300511*x^6 + 4918268*x^7 + 82918049*x^8 + 1430142380*x^9 + 25115651237*x^10 + ...
-
{a(n) = my(A=[1,1,0,0,0,0,0,0,0]); for(i=1, n, A = concat(A, 0);
A[#A-7] = polcoeff( sum(m=-#A, #A, x^m * Ser(A)^m * (1 - x^m +x*O(x^n))^(m+14) ), #A-64)); A[n+1]}
for(n=0, 30, print1(a(n), ", "))
A384828
G.f. A(x) satisfies 1/x^71 = Sum_{n=-oo..+oo} A(x)^n * x^n * (1 - x^n)^(n+16).
Original entry on oeis.org
1, 1, 9, 120, 1839, 30862, 548783, 10160786, 193811734, 3782270289, 75158649892, 1515578476370, 30935212293083, 637920390487505, 13269865608471203, 278121828806207328, 5867506406619195047, 124502776024601555996, 2655381364988431518262, 56892952987400631546208, 1223972213493916563960331
Offset: 0
G.f.: A(x) = 1 + x + 9*x^2 + 120*x^3 + 1839*x^4 + 30862*x^5 + 548783*x^6 + 10160786*x^7 + 193811734*x^8 + 3782270289*x^9 + 75158649892*x^10 + ...
-
{a(n) = my(A=[1,1,0,0,0,0,0,0,0,0]); for(i=1, n, A = concat(A, 0);
A[#A-8] = -polcoeff( sum(m=-#A, #A, x^m * Ser(A)^m * (1 - x^m +x*O(x^n))^(m+16) ), #A-81)); A[n+1]}
for(n=0, 30, print1(a(n), ", "))
A366730
Expansion of g.f. A(x,y) satisfying 0 = Sum_{n=-oo..+oo} x^n * A(x,y)^n * (y - x^(n-1))^(n+1), as a triangle of coefficients T(n,k) of x^n*y^k in A(x,y), read by rows n >= 0.
Original entry on oeis.org
1, 0, 1, 0, -2, 2, 0, 3, -6, 5, 0, -6, 14, -20, 14, 0, 11, -36, 59, -70, 42, 0, -18, 87, -176, 246, -252, 132, 0, 28, -190, 500, -824, 1022, -924, 429, 0, -44, 386, -1312, 2615, -3780, 4236, -3432, 1430, 0, 69, -756, 3218, -7734, 13107, -17112, 17523, -12870, 4862, 0, -104, 1443, -7514, 21496, -42444, 64031, -76692, 72358, -48620, 16796
Offset: 0
G.f.: A(x,y) = 1 + x*y + x^2*(-2*y + 2*y^2) + x^3*(3*y - 6*y^2 + 5*y^3) + x^4*(-6*y + 14*y^2 - 20*y^3 + 14*y^4) + x^5*(11*y - 36*y^2 + 59*y^3 - 70*y^4 + 42*y^5) + x^6*(-18*y + 87*y^2 - 176*y^3 + 246*y^4 - 252*y^5 + 132*y^6) + x^7*(28*y - 190*y^2 + 500*y^3 - 824*y^4 + 1022*y^5 - 924*y^6 + 429*y^7) + x^8*(-44*y + 386*y^2 - 1312*y^3 + 2615*y^4 - 3780*y^5 + 4236*y^6 - 3432*y^7 + 1430*y^8) + x^9*(69*y - 756*y^2 + 3218*y^3 - 7734*y^4 + 13107*y^5 - 17112*y^6 + 17523*y^7 - 12870*y^8 + 4862*y^9) + ...
where A = A(x,y) satisfies
0 = Sum_{n=-oo..+oo} x^n * A^n * (y - x^(n-1))^(n+1);
explicitly,
0 = ((-A + 1)/A)/x + y + (A*y^2 - 2*A*y + ((A^3 - 1)/A^2))*x + A^2*y^3*x^2 + (A^3*y^4 - 3*A^2*y^2)*x^3 + (A^4*y^5 + ((3*A^4 - 1)/A^2)*y)*x^4 + (A^5*y^6 - 4*A^3*y^3 + ((-A^5 + 1)/A^3))*x^5 + A^6*y^7*x^6 + (A^7*y^8 - 5*A^4*y^4 + ((6*A^5 - 1)/A^2)*y^2)*x^7 + A^8*y^9*x^8 + (A^9*y^10 - 6*A^5*y^5 + ((-4*A^6 + 2)/A^3)*y)*x^9 + (A^10*y^11 + ((10*A^6 - 1)/A^2)*y^3)*x^10 + ...
This triangle of coefficients of x^n*y^k in A(x,y) begins:
1;
0, 1;
0, -2, 2;
0, 3, -6, 5;
0, -6, 14, -20, 14;
0, 11, -36, 59, -70, 42;
0, -18, 87, -176, 246, -252, 132;
0, 28, -190, 500, -824, 1022, -924, 429;
0, -44, 386, -1312, 2615, -3780, 4236, -3432, 1430;
0, 69, -756, 3218, -7734, 13107, -17112, 17523, -12870, 4862;
0, -104, 1443, -7514, 21496, -42444, 64031, -76692, 72358, -48620, 16796;
0, 152, -2668, 16862, -56856, 129425, -223458, 307189, -340912, 298298, -184756, 58786;
0, -222, 4782, -36456, 144159, -375618, 734310, -1143924, 1453221, -1504932, 1227876, -705432, 208012; ...
in which the main diagonal equals the Catalan numbers (A000108), and column 1 equals the coefficients in Product_{n>=1} (1 - q^(2*n-1))^2/(1 - q^(2*n))^2 (A274621).
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{T(n,k) = my(A=[1]); for(i=1,n, A = concat(A,0);
A[#A] = polcoeff( sum(n=-#A,#A, x^n * Ser(A)^n * (y - x^(n-1))^(n+1) ), #A-2)); polcoeff(A[n+1],k)}
for(n=0,12, for(k=0,n, print1(T(n,k),", "));print(""))
A366732
Expansion of g.f. A(x) satisfying 0 = Sum_{n=-oo..+oo} x^n * A(x)^n * (2 - x^(n-1))^(n+1).
Original entry on oeis.org
1, 2, 4, 22, 108, 574, 3224, 18592, 109728, 660938, 4041900, 25034000, 156724204, 990127086, 6304425800, 40416596578, 260658078580, 1689976752116, 11008752656960, 72016455973262, 472912945955364, 3116243639293972, 20599091568973324, 136557058462319178, 907668022344460584
Offset: 0
G.f.: A(x) = 1 + 2*x + 4*x^2 + 22*x^3 + 108*x^4 + 574*x^5 + 3224*x^6 + 18592*x^7 + 109728*x^8 + 660938*x^9 + 4041900*x^10 + 25034000*x^11 + ...
-
{a(n) = my(A=[1]); for(i=1,n, A = concat(A,0);
A[#A] = polcoeff( sum(n=-#A,#A, x^n * Ser(A)^n * (2 - x^(n-1))^(n+1) ), #A-2));A[n+1]}
for(n=0,40,print1(a(n),", "))
Showing 1-10 of 14 results.
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