A229041 -id:A229041 - 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

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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

A230218 G.f. A(x) satisfies: [x^n] A(x)^(n^2+n+1) = 0 for n>1.

Original entry on oeis.org

1, 1, -3, 14, -85, 504, -4424, 6796, -878157, -25703710, -1270518018, -65772588300, -3848787714746, -248212765567326, -17520121174143210, -1343050785659060872, -111112550557260635229, -9867409274482580015370, -936234289413196544207234, -94522404087905722536648780

Offset: 0

Paul D. Hanna, Oct 11 2013

sign

			G.f.: A(x) = 1 + x - 3*x^2 + 14*x^3 - 85*x^4 + 504*x^5 - 4424*x^6 +...
Coefficients of x^k in the powers A(x)^(n^2+n+1) of g.f. A(x) begin:
n=0: [1, 1,   -3,    14,    -85,     504,    -4424,      6796, ...];
n=1: [1, 3,   -6,    25,   -153,     819,    -8664,    -18360, ...];
n=2: [1, 7,    0,     7,    -98,     210,   -10122,   -141525, ...];
n=3: [1,13,   39,     0,    -78,    -819,   -15483,   -380952, ...];
n=4: [1,21,  147,   364,      0,   -2457,   -35805,   -821916, ...];
n=5: [1,31,  372,  2139,   5580,       0,   -91698,  -1792947, ...];
n=6: [1,43,  774,  7525,  42097,  125517,        0,  -4097298, ...];
n=7: [1,57, 1425, 20482, 185877, 1089270,  3791298,         0, ...];
n=8: [1,73, 2409, 47450, 619697, 5619978, 35621518, 144591976, 0, ...]; ...
where the coefficients of x^n in A(x)^(n^2+n+1) all equal zero for n>1.
ODD TERMS:
For n>0, a(n) appears to be odd only when n is a power of 2:
a(1) = 1;
a(2) = -3;
a(4) = -85;
a(8) = -878157;
a(16) = -111112550557260635229;
a(32) = -886203693344229341179357569730608605545213045330679133; ...

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

Cf. A185072, A229041, A229044, A171791, A292877.

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

for n>0, a(n) is odd iff n is a power of 2 (conjecture).
G.f. A(x) satisfies:
(1) A(x) = F(x/A(x)) where F(x) = A(x*F(x)) is the g.f. of A185072.
(2) A(x) = G(x/A(x)^2) where G(x) = A(x*G(x)^2) is the g.f. of A229041.
a(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 = 0.015106126717978... - Vaclav Kotesovec, Sep 27 2017

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

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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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A230218 G.f. A(x) satisfies: [x^n] A(x)^(n^2+n+1) = 0 for n>1.

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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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