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A359671
a(n) = coefficient of x^n in A(x) where 1 = Sum_{n=-oo..+oo} (x^n - x*A(x))^n.
2
2, 4, 6, 6, 10, 78, 412, 1394, 3312, 6416, 17454, 83334, 384284, 1377888, 3931286, 10234748, 31776266, 127848076, 527518582, 1910397078, 6035143914, 18202417974, 60151348904, 226355566282, 874920531958, 3166323335574, 10599244540550, 34588365630694, 118339356017608
OFFSET
0,1
COMMENTS
All terms are even: a(n) = 2 * A355868(n) for n >= 0.
Related identity: 0 = Sum_{n=-oo..+oo} x^n * (y - x^n)^n, which holds formally for all y.
LINKS
FORMULA
G.f. A(x) = Sum_{n>=0} a(n)*x^n satisfies:
(1) 1 = Sum_{n=-oo..+oo} (x^n - x*A(x))^n.
(2) 1 = Sum_{n=-oo..+oo} x^(2*n+1) * (x^n + A(x))^n.
(3) 0 = Sum_{n=-oo..+oo} (-1)^n * (x^n - x*A(x))^(n-1).
(4) 0 = Sum_{n=-oo..+oo} (-1)^n * x^(2*n+1) * (x^n + x*A(x))^(n+1).
(5) 1 = Sum_{n=-oo..+oo} x^(n^2) / (1 - A(x)*x^(n+1))^n.
(6) 1 = Sum_{n=-oo..+oo} x^(n^2) / (1 + A(x)*x^(n+1))^(n+1).
(7) 0 = Sum_{n=-oo..+oo} (-1)^n * x^(n*(n-1)) / (1 + A(x)*x^n)^n.
EXAMPLE
G.f.: A(x) = 2 + 4*x + 6*x^2 + 6*x^3 + 10*x^4 + 78*x^5 + 412*x^6 + 1394*x^7 + 3312*x^8 + 6416*x^9 + 17454*x^10 + 83334*x^11 + 384284*x^12 + ...
SPECIFIC VALUES.
A(x) = 3 at x = 0.1794935271005324391410493657541129782265990045275870...
A(x) = 4 at x = 0.2492900841034309263190875287839455698977108450414094...
A(x) = 5 at x = 0.2676600392887397049709560009239544652896107097280049...
PROG
(PARI) {a(n) = my(A=[2]); for(i=1, n, A=concat(A, 0);
A[#A] = polcoeff( sum(m=-#A, #A, (x^m - x*Ser(A))^m ), #A)); A[n+1]}
for(n=0, 30, print1(a(n), ", "))
(PARI) {a(n) = my(A=[2]); for(i=1, n, A=concat(A, 0);
A[#A] = polcoeff( sum(m=-#A, #A, x^(2*m+1) * (x^m + Ser(A))^m ), #A)); A[n+1]}
for(n=0, 30, print1(a(n), ", "))
(PARI) {a(n) = my(A=[2]); for(i=1, n, A=concat(A, 0);
A[#A] = polcoeff( sum(m=-#A, #A, x^(m^2)/(1 - Ser(A)*x^(m+1))^m ), #A)); A[n+1]}
for(n=0, 30, print1(a(n), ", "))
(PARI) {a(n) = my(A=[2]); for(i=1, n, A=concat(A, 0);
A[#A] = polcoeff( sum(m=-#A, #A, x^(m^2)/(1 + Ser(A)*x^(m+1))^(m+1) ), #A)); A[n+1]}
for(n=0, 30, print1(a(n), ", "))
CROSSREFS
Cf. A355868.
Sequence in context: A299541 A066820 A309796 * A222733 A364828 A141677
KEYWORD
nonn
AUTHOR
Paul D. Hanna, Jan 10 2023
STATUS
approved