A366731
Expansion of g.f. A(x) satisfying 0 = Sum_{n=-oo..+oo} x^n * A(x)^n * (1 - x^(n-1))^(n+1).
Original entry on oeis.org
1, 1, 0, 2, 2, 6, 19, 41, 99, 307, 750, 2062, 5776, 15674, 43700, 123729, 345728, 982580, 2801615, 7994268, 22953104, 66128105, 190846074, 552959720, 1605817449, 4673526011, 13635237816, 39860703465, 116739997283, 342538898105, 1006709394181, 2963267980415, 8735388348630
Offset: 0
G.f.: A(x) = 1 + x + 2*x^3 + 2*x^4 + 6*x^5 + 19*x^6 + 41*x^7 + 99*x^8 + 307*x^9 + 750*x^10 + 2062*x^11 + 5776*x^12 + 15674*x^13 + 43700*x^14 + ...
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{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 * (1 - x^(n-1))^(n+1) ), #A-2));A[n+1]}
for(n=0,40,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 + ...
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{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),", "))
A366733
Expansion of g.f. A(x) satisfying 0 = Sum_{n=-oo..+oo} x^n * A(x)^n * (3 - x^(n-1))^(n+1).
Original entry on oeis.org
1, 3, 12, 90, 702, 5838, 50895, 458103, 4225683, 39745665, 379730658, 3674980518, 35951809104, 354950991006, 3532167377340, 35390917028619, 356742401734236, 3615164398809324, 36809446799831823, 376387507560832992, 3863438843523528636, 39794189982905311407
Offset: 0
G.f.: A(x) = 1 + 3*x + 12*x^2 + 90*x^3 + 702*x^4 + 5838*x^5 + 50895*x^6 + 458103*x^7 + 4225683*x^8 + 39745665*x^9 + 379730658*x^10 + ...
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{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 * (3 - x^(n-1))^(n+1) ), #A-2));A[n+1]}
for(n=0,30,print1(a(n),", "))
A366734
Expansion of g.f. A(x) satisfying 0 = Sum_{n=-oo..+oo} x^n * A(x)^n * (4 - x^(n-1))^(n+1).
Original entry on oeis.org
1, 4, 24, 236, 2504, 28332, 335656, 4108688, 51558000, 659737684, 8575826448, 112927383328, 1503232394344, 20195196226124, 273467339844368, 3728623506924660, 51145851271818536, 705322823588365592, 9772995790887474920, 135992755093954566300, 1899633478390401668072
Offset: 0
G.f.: A(x) = 1 + 4*x + 24*x^2 + 236*x^3 + 2504*x^4 + 28332*x^5 + 335656*x^6 + 4108688*x^7 + 51558000*x^8 + 659737684*x^9 + 8575826448*x^10 + ...
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{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 * (4 - x^(n-1))^(n+1) ), #A-2));A[n+1]}
for(n=0,30,print1(a(n),", "))
Original entry on oeis.org
1, -2, 14, -176, 2615, -42444, 734310, -13332898, 251087228, -4863520344, 96340129818, -1943639738074, 39815238143374, -826201916477272, 17334983283537509, -367213838120451038, 7844257467257818627, -168807941163188191336, 3656662240133060807499, -79675906058698383705100
Offset: 0
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{A366730(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=0,20, print1(A366730(2*n,n),", "))
Showing 1-6 of 6 results.
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