A230218 -id:A230218 - cp's OEIS frontend

A171791 G.f. A(x) satisfies: [x^n] A(x)^((n+1)^2) = 0 for n>1 with a(0)=a(1)=1.

1, 1, -4, 25, -194, 1603, -15264, 122316, -1897710, -8845133, -1169435932, -52853978047, -3193246498792, -205347570309000, -14534295599537024, -1115833257773950536, -92445637289048967654, -8219735646409095418617

Offset: 0

Paul D. Hanna, Jan 24 2010

sign

It appears that for k>0, a(k) is odd iff k = 2*A003714(n)+1 for n>=0, where A003714 is the fibbinary numbers (integers whose binary representation contains no consecutive ones); this is true for at least the first 520 terms. [See also A263190 and A263075.] - Paul D. Hanna, Oct 09 2013

Observation of Paul D. Hanna is true for at least the first 1028 terms. - Sean A. Irvine, Apr 25 2014

			G.f.: A(x) = 1 + x - 4*x^2 + 25*x^3 - 194*x^4 + 1603*x^5 +...
The coefficients in the square powers of g.f. A(x) begin:
A^1:  [1,  1,   -4,    25,   -194,    1603,   -15264,    122316, ...];
A^4:  [1,  4,  -10,    56,   -427,    3360,   -33546,    218880, ...];
A^9:  [1,  9,    0,    21,   -252,    1701,   -25992,     -2970, ...];
A^16: [1, 16,   56,     0,    -84,    -784,   -18656,   -384896, ...];
A^25: [1, 25,  200,   525,      0,   -2695,   -38600,   -878150, ...];
A^36: [1, 36,  486,  3000,   7821,       0,  -101322,  -1916352, ...];
A^49: [1, 49,  980, 10241,  58898,  170079,        0,  -4515000, ...];
A^64: [1, 64, 1760, 27136, 256048, 1500352,  4979712,         0, ...];
A^81: [1, 81, 2916, 61425, 838026, 7720839, 48097152, 184870512, 0,...]; ...
Note how the coefficient of x^n in A(x)^((n+1)^2) = 0 for n>1.
ALTERNATE RELATION.
The coefficients in A(x)^(n^2) * (1 - n*x*A(x)'/A(x)) begin:
n=1: [1, 0, 4, -50, 582, -6412, 76320, -733896, 13283970, ...];
n=2: [1, 2, 0, -28, 427, -5040, 67092, -547200, 15539502, ...];
n=3: [1, 6, 0, 0, 84, -1134, 25992, 3960, 13172355, ...];
n=4: [1, 12, 28, 0, 0, 196, 9328, 288672, 13426530, ...];
n=5: [1, 20, 120, 210, 0, 0, 7720, 351260, 15775425, ...];
n=6: [1, 30, 324, 1500, 2607, 0, 0, 319392, 17452530, ...];
n=7: [1, 42, 700, 5852, 25242, 48594, 0, 0, 15518020, ...];
n=8: [1, 56, 1320, 16960, 128024, 562632, 1244928, 0, 0, ...];
n=9: [1, 72, 2268, 40950, 465570, 3431484, 16032384, 41082336, 0, 0, ...]; ...
in which the two adjacent diagonals above the main diagonal are all zeros after initial terms, illustrating that
(1) 0 = [x^(n-1)] A(x)^(n^2) * (1 - n*x*A(x)'/A(x)), and
(2) 0 = [x^n] A(x)^(n^2) * (1 - n*x*A(x)'/A(x)), for n > 0.

Paul D. Hanna, Table of n, a(n) for n = 0..200

Cf. A263190, A263075, A229044, A185072, A229041, A230218, A292877.

{a(n) = local(A=[1,1]); for(m=3,n+1, A=concat(A,0); A[ #A]=-Vec(Ser(A)^(m^2))[m]/m^2); A[n+1]}
for(n=0, 20, print1(a(n), ", "))

The g.f. A(x) satisfies the following relations.
(1) 0 = [x^(n-1)] A(x)^(n^2), for n > 1.
(2) 0 = [x^(n-1)] A(x)^(n^2) * (1 - n*x*A(x)'/A(x)), for n > 1. - Paul D. Hanna, Oct 22 2020
(3) 0 = [x^n] A(x)^(n^2) * (1 - n*x*A(x)'/A(x)), for n > 0. - Paul D. Hanna, Oct 22 2020

A185072 G.f.: A(x) satisfies: [x^n] A(x)^(n^2-2*n+2) = 0 for n>=2.

Original entry on oeis.org

1, 1, -2, 6, -28, 70, -1446, -22302, -855032, -33435486, -1541257250, -80299386706, -4675923739764, -300809006015466, -21184444811987030, -1620869900459370150, -133878027649528854000, -11872222666784936265342, -1125045987661214982721602, -113458738692543731877937418

Offset: 0

Paul D. Hanna, Jan 22 2012

sign

It appears that (n+1) divides [x^n] A(x)^2 for n>=0 (A229128).

			G.f.: A(x) = 1 + x - 2*x^2 + 6*x^3 - 28*x^4 + 70*x^5 - 1446*x^6 -...
Coefficients of x^k in the powers A(x)^(n^2-2*n+2) of g.f. A(x) begin:
n=1: [1,  1,   -2,     6,    -28,     70,   -1446,   -22302, ...];
n=2: [1,  2,   -3,     8,    -40,     60,   -2604,   -48112, ...];
n=3: [1,  5,    0,     0,    -35,   -189,   -5760,  -140700, ...];
n=4: [1, 10,   25,     0,    -70,   -728,  -13410,  -339000, ...];
n=5: [1, 17,  102,   238,      0,  -2142,  -32198,  -743886, ...];
n=6: [1, 26,  273,  1456,   3822,      0,  -80366, -1638312, ...];
n=7: [1, 37,  592,  5328,  29045,  89947,       0, -3630588, ...];
n=8: [1, 50, 1125, 15000, 130900, 769860, 2823600,        0, ...]; ...
where the coefficients of x^n in A(x)^(n^2-2*n+2) all equal zero for n>1.
RELATED FUNCTIONS.
The coefficients in G_n(x) that satisfy G_n(x) = A(x*G_n(x)^n) begin:
G_1: [1, 1,-1,  1,  -7,   -49, -1191, -31569,-1051695, -41520593, ...];
G_2: [1, 1, 0, -1,  -6,   -78, -1544, -40605,-1328178, -51857806, ...];
G_3: [1, 1, 1,  0,  -9,  -117, -2118, -53232,-1699905, -65386779, ...];
G_4: [1, 1, 2,  4,   0,  -141, -2958, -71900,-2216860, -83454920, ...];
G_5: [1, 1, 3, 11,  37,     0, -3245, -95286,-2941059,-108180433, ...];
G_6: [1, 1, 4, 21, 118,   581,     0, -99086,-3760182,-141280086, ...];
G_7: [1, 1, 5, 34, 259,  2002, 13212,      0,-3775221,-176047295, ...];
G_8: [1, 1, 6, 50, 476,  4788, 47578, 397090,       0,-172383145, ...];
G_9: [1, 1, 7, 69, 785,  9589,120333,1468749,14889577,         0, ...];
G_10:[1, 1, 8, 91,1202, 17180,256056,3859425,56018694, 669865615, 0, ...]; ...
Note how that the coefficients of x^n in G_n(x) are zero for n>1.

Paul D. Hanna, Table of n, a(n) for n = 0..200

Cf. A185082, A229041, A229044, A229128, A230218.

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

G.f. A(x) satisfies:
(1) A(x) = F(x/A(x)) where F(x) = A(x*F(x)) is the g.f. of A229041.
(2) A(x) = G(x/A(x)^2) where G(x) = A(x*G(x)^2) is the g.f. of A229044.
(3) A(x) = H(x*A(x)) where H(x) = A(x/H(x)) is the g.f. of A230218.
(4) [x^n] G_n(x) = 0 for n>1 where G_n(x) = A( x*G_n(x)^n ) and A(x) = G_n( x/A(x)^n ).

Name changed and entry revised by Paul D. Hanna, Oct 11 2013

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

Original entry on oeis.org

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

A229041 G.f. A(x) satisfies: [x^n] A(x)^(n^2-n+1) = 0 for n>=2.

Original entry on oeis.org

1, 1, -1, 1, -7, -49, -1191, -31569, -1051695, -41520593, -1896894223, -98362962257, -5705059841823, -365846227736001, -25696840682622175, -1961769357361345473, -161728572333727674687, -14318505129615014956737, -1354916705432679538845759, -136467389971873491004759617

Offset: 0

Paul D. Hanna, Sep 14 2013

sign

G.f. A(x) satisfies:
(1) A(x) = G(x*A(x)) where G(x) = A(x/G(x)) is the g.f. of A185072.
(2) A(x) = F(x/A(x)) where F(x) = A(x*F(x)) is the g.f. of A229044.

			G.f.: A(x) = 1 + x - x^2 + x^3 - 7*x^4 - 49*x^5 - 1191*x^6 - 31569*x^7 -...
Coefficients of x^k in the powers A(x)^(n^2-n+1) of g.f. A(x) begin:
n=1: [1, 1,  -1,    1,    -7,    -49,   -1191,   -31569,  -1051695, ...];
n=2: [1, 3,   0,   -2,   -15,   -189,   -3850,  -101700,  -3340845, ...];
n=3: [1, 7,  14,    0,   -56,   -588,  -10808,  -273972,  -8760325, ...];
n=4: [1,13,  65,  143,     0,  -1742,  -27534,  -638690, -19496334, ...];
n=5: [1,21, 189,  931,  2478,      0,  -67312, -1444608, -40653711, ...];
n=6: [1,31, 434, 3596, 19158,  62062,       0, -3116120, -84939504, ...];
n=7: [1,43, 860,10578, 88795, 526449, 2045854,        0,-167991196, ...];
n=8: [1,57,1539,26125,311619,2754297,18283187, 83718693,         0, ...];
...
where the coefficients of x^n in A(x)^(n^2-n+1) all equal zero for n>=2.

Cf. A185072, A229044, A230218, A171791.

{a(n)=local(A=[1,1]);for(k=1,n,A=concat(A,0);A[#A]=-polcoeff((Ser(A) +O(x^(k+2)))^(k^2+k+1)/(k^2+k+1),k+1));A[n+1]}
for(n=0,30,print1(a(n),", "))

A229044 G.f. A(x) satisfies: [x^(n+1)] A(x)^(n^2) = 0 for n>=0.

Original entry on oeis.org

1, 1, 0, -1, -6, -78, -1544, -40605, -1328178, -51857806, -2350025232, -121120896906, -6991877399100, -446673990116508, -31277285155060464, -2381645560450404989, -195914136385421694954, -17312472044077536945630, -1635541992950202705979424, -164494265246550280147797438

Offset: 0

Paul D. Hanna, Sep 12 2013

sign

			G.f.: A(x) = 1 + x - x^3 - 6*x^4 - 78*x^5 - 1544*x^6 - 40605*x^7 -...
Coefficients of x^k in the square powers A(x)^(n^2) of g.f. A(x) begin:
n=1: [1, 1,   0,   -1,    -6,    -78,   -1544,   -40605,  -1328178, ...];
n=2: [1, 4,   6,    0,   -35,   -396,   -7182,  -181824,  -5817510, ...];
n=3: [1, 9,  36,   75,     0,  -1260,  -21408,  -499203, -15299145, ...];
n=4: [1,16, 120,  544,  1484,      0,  -52656, -1202240, -34269906, ...];
n=5: [1,25, 300, 2275, 11900,  40680,       0, -2557775, -73526475, ...];
n=6: [1,36, 630, 7104, 57429, 345204, 1430418,        0,-142432290, ...];
n=7: [1,49,1176,18375,209230,1833678,12546744, 61418175,         0, ...];
n=8: [1,64,2016,41600,630960,7470336,71271616,549420288,3113335320, 0, ...]; ...
where the coefficients of x^(n+1) in A(x)^(n^2) all equal zero for n>=0.
Related expansions.
A(x) = G(x*A(x)^2) where G(x) = A(x/G(x)^2) is the g.f. of A185072:
G(x) = 1 + x - 2*x^2 + 6*x^3 - 28*x^4 + 70*x^5 - 1446*x^6 -...
A(x)'/A(x) = 1 - x - 2*x^2 - 21*x^3 - 364*x^4 - 8830*x^5 - 273972*x^6 - 10313037*x^7 - 455135384*x^8 - 22995056286*x^9 - 1307053358940*x^10 - ...
A(x)/A(x)' = 1 + x + 3*x^2 + 26*x^3 + 417*x^4 + 9726*x^5 + 295000*x^6 + 10946172*x^7 + 478392123*x^8 + ... + A305144(n)*x^n + ...

Cf. A185072, A305144, A230218, A229041, A171791.

{a(n)=local(A=[1,1]);for(k=1,n,A=concat(A,0);A[#A]=-polcoeff((Ser(A) +O(x^(k+2)))^(k^2)/(k^2),k+1));A[n+1]}
for(n=0,30,print1(a(n),", "))

a(n) is odd iff n+1 is a power of 2 (conjecture).
G.f. A(x) satisfies the following relationes.
(1) [x^(n+1)] A(x)^(n^2) = 0 for n>=0.
(2) A(x) = G(x*A(x)^2) where G(x) = A(x/G(x)^2) is the g.f. of A185072.
(3) A(x)/A(x)' is the g.f. of A305144. - Paul D. Hanna, Oct 23 2020

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

A171791 G.f. A(x) satisfies: [x^n] A(x)^((n+1)^2) = 0 for n>1 with a(0)=a(1)=1.

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A185072 G.f.: A(x) satisfies: [x^n] A(x)^(n^2-2*n+2) = 0 for n>=2.

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

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A229041 G.f. A(x) satisfies: [x^n] A(x)^(n^2-n+1) = 0 for n>=2.

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A229044 G.f. A(x) satisfies: [x^(n+1)] A(x)^(n^2) = 0 for n>=0.

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

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