A303562 -id:A303562 - cp's OEIS frontend

A292877 G.f. A(x) satisfies: [x^(n-1)] 1/A(x)^(n^2) = 0 for n>2.

1, 1, 5, 34, 273, 2331, 22110, 190450, 2540975, -1071509, 1200284739, -50263360280, 3102388877800, -199436004737160, 14155468007742978, -1088800915851203694, 90359645776680747647, -8046100605226675723225, 765244962799789283768523, -77422876485545489461403294, 8303247917673506082303329493, -940940782152450052071048090369, 112352003582903383388702940258120

Offset: 0

Paul D. Hanna, Sep 25 2017

sign

Conjectures:
(1) a(2^n) is odd for n>=0.
(2) a(n) is odd iff n is a Fibbinary number: a(A003714(k)) is odd for k>=0.
(3) The number of odd terms between a(2^n) and a(2^(n+1)-1), inclusively, is Fibonacci(n+1), for n>=0.

			G.f.: A(x) = 1 + x + 5*x^2 + 34*x^3 + 273*x^4 + 2331*x^5 + 22110*x^6 + 190450*x^7 + 2540975*x^8 - 1071509*x^9 + 1200284739*x^10 - 50263360280*x^11 + 3102388877800*x^12 - 199436004737160*x^13 + 14155468007742978*x^14 - 1088800915851203694*x^15 +...
such that the coefficient of x^n in 1/A(x)^(n^2) equals zero for n>1.
Notice that a(n) seems to be odd only when n is a Fibbinary number (A003714):
[0, 1, 2, 4, 5, 8, 9, 10, 16, 17, 18, 20, 21, 32, 33, 34, 36, 37, 40, 41, 42, 64, 65, 66, 68, 69, 72, 73, 74, 80, 81, 82, 84, 85, 128, 29, 130, 132, 133, ...].
RELATED TABLES.
(1) The table of coefficients in 1/A(x)^(n^2) begins:
n=1: [1, -1, -4, -25, -194, -1603, -15264, -122316, -1897710, ...];
n=2: [1, -4, -10, -56, -427, -3360, -33546, -218880, -5179834, ...];
n=3: [1, -9, 0, -21, -252, -1701, -25992, 2970, -7903413, ...];
n=4: [1, -16, 56, 0, -84, 784, -18656, 384896, -13426530, ...];
n=5: [1, -25, 200, -525, 0, 2695, -38600, 878150, -26292375, ...];
n=6: [1, -36, 486, -3000, 7821, 0, -101322, 1916352, -52357590, ...];
n=7: [1, -49, 980, -10241, 58898, -170079, 0, 4515000, -108626140, ...];
n=8: [1, -64, 1760, -27136, 256048, -1500352, 4979712, 0, -234893352, ...];
n=9: [1, -81, 2916, -61425, 838026, -7720839, 48097152, -184870512, 0, ...]; ...
such that the main diagonal is all zeros after the initial terms.
(2) The table of coefficients in (1/x)*Series_Reversion( x*A(x)^n ) begins:
n=1: [1, -1, -3, -14, -85, -504, -4424, -6796, -878157, ...];
n=2: [1, -2, -3, -8, -40, -60, -2604, 48112, -1747260, ...];
n=3: [1, -3, 0, 2, -15, 189, -3850, 101700, -3340845, ...];
n=4: [1, -4, 6, 0, -35, 396, -7182, 181824, -5817510, ...];
n=5: [1, -5, 15, -30, 0, 714, -13335, 315060, -9679455, ...];
n=6: [1, -6, 27, -104, 315, 0, -19957, 532848, -15864336, ...];
n=7: [1, -7, 42, -238, 1260, -5481, 0, 713796, -25010433, ...];
n=8: [1, -8, 60, -448, 3310, -23352, 136696, 0, -31112163, ...];
n=9: [1, -9, 81, -750, 7065, -66420, 598626, -4474764, 0, ...]; ...
in which the main diagonal is all zeroes after the initial terms.
(3) The table of coefficients in ((1/x)*Series_Reversion( x*A(x)^n ))^(1/n) begins:
n=1: [1, -1, -3, -14, -85, -504, -4424, -6796, -878157, ...];
n=2: [1, -1, -2, -6, -28, -70, -1446, 22302, -855032, ...];
n=3: [1, -1, -1, -1, -7, 49, -1191, 31569, -1051695, ...];
n=4: [1, -1, 0, 1, -6, 78, -1544, 40605, -1328178, ...];
n=5: [1, -1, 1, 0, -9, 117, -2118, 53232, -1699905, ...];
n=6: [1, -1, 2, -4, 0, 141, -2958, 71900, -2216860, ...];
n=7: [1, -1, 3, -11, 37, 0, -3245, 95286, -2941059, ...];
n=8: [1, -1, 4, -21, 118, -581, 0, 99086, -3760182, ...];
n=9: [1, -1, 5, -34, 259, -2002, 13212, 0, -3775221, ...];
n=10: [1, -1, 6, -50, 476, -4788, 47578, -397090, 0, ...]; ...
in which the secondary diagonal is all zeroes after the initial terms.
(4) The table of coefficients in 1/A(x)^n begins:
n=1: [1, -1, -4, -25, -194, -1603, -15264, -122316, ...];
n=2: [1, -2, -7, -42, -322, -2618, -25145, -191580, ...];
n=3: [1, -3, -9, -52, -396, -3168, -30889, -220332, ...];
n=4: [1, -4, -10, -56, -427, -3360, -33546, -218880, ...];
n=5: [1, -5, -10, -55, -425, -3286, -33990, -195585, ...];
n=6: [1, -6, -9, -50, -399, -3024, -32938, -157122, ...];
n=7: [1, -7, -7, -42, -357, -2639, -30968, -108718, ...];
n=8: [1, -8, -4, -32, -306, -2184, -28536, -54368, ...];
n=9: [1, -9, 0, -21, -252, -1701, -25992, 2970,  -7903413, ...]; ...
where the main diagonal divided by n begins:
D = [1, -2/2, -9/3, -56/4, -425/5, -3024/6, -30968/7, -54368/8, -7903413/9, ...],
D = [1, -1, -3, -14, -85, -504, -4424, -6796, -878157, 25703710, -1270518018, ...].
Compare D to:
A230218 = [1, 1, -3, 14, -85, 504, -4424, 6796, -878157, -25703710, -1270518018, ...];
the g.f. G(x) of A230218 obeys: [x^n] G(x)^(n^2+n+1) = 0 for n>1.

Paul D. Hanna, Table of n, a(n) for n = 0..520

Cf. A303562, A003714, A230218, A171791.

{a(n) = my(A=[1, 1]); for(i=2, n+1, A=concat(A, 0); A[#A]=Vec(1/Ser(A)^((#A)^2))[#A]/(#A)^2 ); A[n+1]}
for(n=0, 30, print1(a(n), ", "))

G.f. A(x) satisfies:
(1) [x^(n-1)] 1/A(x)^(n^2) = 0 for n>2.
(2) [x^(n-1)] (1/x)*Series_Reversion( x*A(x)^n ) = 0 for n>2.
(3) [x^(n-2)] ( (1/x)*Series_Reversion( x*A(x)^n ) )^(1/n) = 0 for n>3.
a(n) ~ (-1)^n * c * 2^(2*n) * n^(n-5/2) / (exp(n) * d^n * (2-d)^n), where d = -LambertW(-2*exp(-2)) = -A226775 = 0.40637573995995990767695812412483975821... and c = 0.01284812446900190... - Vaclav Kotesovec, Sep 27 2017

A303563 G.f. A(x) satisfies: 0 = [x^(n-1)] 1 / A(x)^(n^2-n+1) for n>2.

Original entry on oeis.org

1, 1, 4, 21, 132, 840, 6798, 26187, 982794, -23411010, 1229958800, -63402693620, 3727765284702, -241049598495378, 17055417754898346, -1310070866036785677, 108572840286328367574, -9656468474317765916970, 917437511140569561151848, -92733586081750860360411954, 9936829948115042380890921976, -1125196473407637775842431681496, 134258639558360961220987962351588

Offset: 0

Paul D. Hanna, Apr 27 2018

sign

a(n) is odd iff n = 2^k - 1 for k >= 0 (conjecture).

			G.f.: A(x) = 1 + x + 4*x^2 + 21*x^3 + 132*x^4 + 840*x^5 + 6798*x^6 + 26187*x^7 + 982794*x^8 - 23411010*x^9 + 1229958800*x^10 + ...
such that the coefficient of x^(n-1) in 1/A(x)^(n^2-n+1) equals zero for n>2.
RELATED SERIES.
1/A(x) = 1 - x - 3*x^2 - 14*x^3 - 85*x^4 - 504*x^5 - 4424*x^6 - 6796*x^7 - 878157*x^8 + ... + (-1)^n*A230218(n)*x^n + ...
A'(x)/A(x) = 1 + 7*x + 52*x^2 + 427*x^3 + 3286*x^4 + 32938*x^5 + 108718*x^6 + 7379059*x^7 - 221516750*x^8 + 12494464362*x^9 - 710385713212*x^10 + ...
ILLUSTRATION OF DEFINITION.
The table of coefficients of x^k in 1/A(x)^(n^2-n+1) begins:
n=1: [1, -1, -3, -14, -85, -504, -4424, -6796, -878157, ...];
n=2: [1, -3, -6, -25, -153, -819, -8664, 18360, ...];
n=3: [1, -7, 0, -7, -98, -210, -10122, 141525, ...];
n=4: [1, -13, 39, 0, -78, 819, -15483, 380952, ...];
n=5: [1, -21, 147, -364, 0, 2457, -35805, 821916, ...];
n=6: [1, -31, 372, -2139, 5580, 0, -91698, 1792947, ...];
n=7: [1, -43, 774, -7525, 42097, -125517, 0, 4097298, ...];
n=8: [1, -57, 1425, -20482, 185877, -1089270, 3791298, 0, ...]; ...
in which the main diagonal equals zeros after the initial terms, illustrating that 0 = [x^(n-1)] 1/A(x)^(n^2-n+1) for n>2.
RELATED TABLES.
The table of coefficients of x^k in (x*A(x)^n)' / A(x)^(n^2+1) begins:
n=1: [1, 0, 3, 28, 255, 2016, 22120, 40776, ...];
n=2: [1, -1, 2, 25, 255, 1911, 25992, -67320, ...];
n=3: [1, -4, 0, 2, 70, 240, 15906, -283050, ...];
n=4: [1, -9, 15, 0, 18, -441, 13101, -439560, ...];
n=5: [1, -16, 77, -104, 0, -468, 15345, -547944, ...];
n=6: [1, -25, 228, -897, 1260, 0, 14790, -636207, ...];
n=7: [1, -36, 522, -3850, 14685, -23352, 0, -571716, ...];
n=8: [1, -49, 1025, -11858, 81525, -324870, 598626, 0,  ...]; ...
in which the main diagonal equals zeros after the initial terms, illustrating that 0 = [x^(n-1)] (x*A(x)^n)' / A(x)^(n^2+1) for n>2.
The table of coefficients of x^k in (x*A(x)^n)' / A(x)^(n^2) begins:
n=1: [1, 1, 7, 52, 427, 3286, 32938, 108718, ...];
n=2: [1, 0, 5, 44, 399, 3016, 35670, -3960, ...];
n=3: [1, -3, 0, 7, 120, 630, 19906, -265455, ...];
n=4: [1, -8, 10, 0, 21, -456, 13950, -450240, ...];
n=5: [1, -15, 65, -70, 0, -539, 16215, -566100, ...];
n=6: [1, -24, 207, -748, 882, 0, 16887, -665604, ...];
n=7: [1, -35, 490, -3451, 12299, -17017, 0, -645000, ...];
n=8: [1, -48, 980, -11008, 72870, -274880, 451776, 0, ...]; ...
in which the main diagonal equals zeros after the initial term, illustrating that 0 = [x^(n-1)] (x*A(x)^n)' / A(x)^(n^2) for n>1.

Paul D. Hanna, Table of n, a(n) for n = 0..400

Cf. A303562, A230218, A292877.

{a(n) = my(A=[1, 1]); for(i=1, n, A=concat(A, 0); m=#A; A[m] = Vec( (x*Ser(A)^m)'/Ser(A)^(m^2+1) )[m] ); A[n+1]}
for(n=0,30,print1(a(n),", "))

G.f. A(x) satisfies:
(1) 0 = [x^(n-1)] 1 / A(x)^(n^2-n+1) for n>2.
(2) 0 = [x^(n-1)] (x*A(x))' / A(x)^(n^2-n+2) for n>2.
(3) 0 = [x^(n-1)] (x*A(x)^n)' / A(x)^(n^2+1) for n>2.
(4) 0 = [x^(n-1)] (x*A(x)^(n-1))' / A(x)^(n^2) for n>2.
(5) 0 = [x^(n-1)] (x*A(x)^n)' / A(x)^(n^2) for n>1.

cp's OEIS Frontend

A292877 G.f. A(x) satisfies: [x^(n-1)] 1/A(x)^(n^2) = 0 for n>2.

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