A263190 -id:A263190 - 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

A118113 Even Fibbinary numbers + 1; also 2*Fibbinary(n) + 1.

Original entry on oeis.org

1, 3, 5, 9, 11, 17, 19, 21, 33, 35, 37, 41, 43, 65, 67, 69, 73, 75, 81, 83, 85, 129, 131, 133, 137, 139, 145, 147, 149, 161, 163, 165, 169, 171, 257, 259, 261, 265, 267, 273, 275, 277, 289, 291, 293, 297, 299, 321, 323, 325, 329, 331, 337, 339, 341, 513, 515, 517

Offset: 0

Labos Elemer, Apr 13 2006

m for which binomial(3*m-2,m) (see A117671) is odd, since by Kummer's theorem that happens exactly when the binary expansions of m and 2*m-2 have no 1 bit at the same position in each, and so m odd and no 11 bit pairs except optionally the least significant 2 bits. - Kevin Ryde, Jun 14 2025

Alois P. Heinz, Table of n, a(n) for n = 0..1000

Cf. A000108, A118112, A022341.
Cf. A003714 (Fibbinary numbers), A022340 (even Fibbinary numbers).
Cf. A117671, A263190, A171791, A263075.

F:= combinat[fibonacci]:
b:= proc(n) local j;
      if n=0 then 0
    else for j from 2 while F(j+1)<=n do od;
         b(n-F(j))+2^(j-2)
      fi
    end:
a:= n-> 2*b(n)+1:
seq(a(n), n=0..70);  # Alois P. Heinz, Aug 03 2012

Select[Table[Mod[Binomial[3*k,k], k+1], {k,1200}], #>0&]

a(n) = A022340(n) + 1.
a(n) = 2*A003714(n) + 1.
Solutions to {x : binomial(3x,x) mod (x+1) != 0 } are given in A022341. The corresponding values of binomial(3x,x) mod (x+1) are given here.

New definition from T. D. Noe, Dec 19 2006

A263075 G.f. satisfies: [x^(n-1)] A(x)^(n^2) = n^n * (n-1)! for n>=1.

Original entry on oeis.org

1, 1, 2, 31, 1150, 68713, 5914776, 692005074, 105932315154, 20617891510063, 4984425649932314, 1467604324373250545, 517561005098562714944, 215501019188749426210440, 104642607303457024105207408, 58625315029802441203026824094, 37541542090285460025870424920666

Offset: 0

Paul D. Hanna, Oct 08 2015

nonn

			G.f.: A(x) = 1 + x + 2*x^2 + 31*x^3 + 1150*x^4 + 68713*x^5 + 5914776*x^6 + 692005074*x^7 + 105932315154*x^8 +...
The coefficients in A(x)^(n^2) begin:
n=1: [1, 1, 2, 31, 1150, 68713, 5914776, 692005074, 105932315154, ...];
n=2: [1, 4, 14, 152, 5021, 289824, 24532494, 2841972672, 432284291486, ...];
n=3: [1, 9, 54, 507, 13356, 715635, 58722228, 6685822296, 1005887241243, ...];
n=4: [1, 16, 152, 1536, 31500, 1468016, 114260704, 12668897920, ...];
n=5: [1, 25, 350, 4275, 75000, 2840855, 202155100, 21547156900, ...];
n=6: [1, 36, 702, 10776, 184725, 5598720, 344795598, 34598389248, ...];
n=7: [1, 49, 1274, 24647, 456386, 11753973, 592950960, 54103596918, ...];
n=8: [1, 64, 2144, 51712, 1092016, 26366656, 1071635712, 84557168640, ...];
n=9: [1, 81, 3402, 100791, 2482650, 61309629, 2096140032, 135856780686, ...]; ...
where the terms along the main diagonal begin:
[1, 4, 54, 1536, 75000, 5598720, 592950960, 84557168640, ..., n^n*(n-1)!, ...].
Note that odd terms a(n) occur at positions n starting with:
[0, 1, 3, 5, 9, 11, 17, 19, 21, 33, 35, 37, 41, 43, 65, 67, 69, 73, 75, 81, 83, 85, 129, 131, 133, 137, 139, 145, 147, 149, 161, 163, 165, 169, 171, 257, ...],
which seems to equal A118113, the even Fibbinary numbers + 1, with an initial zero included.

Paul D. Hanna, Table of n, a(n) for n = 0..300

Cf. A171791, A263190, A003714, A118113, A244589.

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

a(n) ~ exp(1-exp(-1)) * n! * n^(n-1). - Vaclav Kotesovec, Oct 20 2020

A294360 G.f. A(x) satisfies: [x^(n-1)] A(x)^(n^2) = (n^2)^(n-1) for n>=1.

Original entry on oeis.org

1, 1, 5, 146, 9935, 1161399, 206499453, 52093726159, 17770811461875, 7903030237890371, 4450363873663943294, 3098938855124650814264, 2616552190721485829559668, 2635178871851323631797948230, 3121810359776427044817295874677, 4298670834657263815567279951080956, 6809336162211769799756516349665301635, 12296952422064277377043754761717448273557, 25116528778581121454413639996325045161219974

Offset: 0

Paul D. Hanna, Nov 01 2017

nonn

			G.f.: A(x) = 1 + x + 5*x^2 + 146*x^3 + 9935*x^4 + 1161399*x^5 + 206499453*x^6 + 52093726159*x^7 + 17770811461875*x^8 + 7903030237890371*x^9 + 4450363873663943294*x^10 + 3098938855124650814264*x^11 + 2616552190721485829559668*x^12 +...
such that the coefficient of x^(n-1) in A(x)^(n^2) equals (n^2)^(n-1) for n>=1.
The table of coefficients of x^k in A(x)^(n^2) begin:
n=1: [1, 1, 5, 146, 9935, 1161399, 206499453, ...];
n=2: [1, 4, 26, 648, 41703, 4775648, 840796570, ...];
n=3: [1, 9, 81, 1758, 102213, 11266209, 1949437539, ...];
n=4: [1, 16, 200, 4096, 207220, 21470032, 3617873616, ...];
n=5: [1, 25, 425, 8950, 390625, 36920005, 5985228975, ...];
n=6: [1, 36, 810, 18696, 723375, 60466176, 9272944890, ...];
n=7: [1, 49, 1421, 37338, 1347843, 97588547, 13841287201, ...];
n=8: [1, 64, 2336, 71168, 2535248, 159036480, 20303433408, 4398046511104, ...]; ...
in which the main diagonal begins:
[1, 4, 81, 4096, 390625, 60466176, 13841287201, 4398046511104, ..., (n^2)^(n-1), ...].

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

Cf. A263190, A263075, A171791, A068087.

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

a(n) ~ c * n^(2*n - 2), where c = exp(2 - exp(-2)) = 6.453771681742981632532303... - Vaclav Kotesovec, Aug 11 2021, updated Mar 18 2024

A371673 Expansion of g.f. A(x) satisfying [x^(n-1)] A(x)^(n^2) = A000108(n-1) * n^n for n >= 1, where A000108 is the Catalan numbers.

Original entry on oeis.org

1, 1, 2, 15, 284, 8575, 345460, 17190684, 1012901520, 68810750943, 5291667341342, 454479660308531, 43140290728900554, 4487833959824527910, 508072065566891421336, 62222074620010689986918, 8200304581300850453687880, 1157674985567876068399895997, 174357014524193551292388873190

Offset: 0

Paul D. Hanna, Apr 02 2024

nonn

Conjecture: a(n) is odd for n > 0 iff n = 2*A003714(k) + 1 for some k, where A003714 is the Fibbinary numbers (integers whose binary representation contains no consecutive ones). See A263075, A263190, and A171791.

			G.f.: A(x) = 1 + x + 2*x^2 + 15*x^3 + 284*x^4 + 8575*x^5 + 345460*x^6 + 17190684*x^7 + 1012901520*x^8 + 68810750943*x^9 + 5291667341342*x^10 + ...
The table of coefficients of x^k in A(x)^(n^2) begin:
 n=1: [1,  1,    2,    15,    284,    8575,    345460, ...];
 n=2: [1,  4,   14,    88,   1365,   38304,   1497150, ...];
 n=3: [1,  9,   54,   363,   4410,  105705,   3874824, ...];
 n=4: [1, 16,  152,  1280,  13804,  263408,   8535648, ...];
 n=5: [1, 25,  350,  3875,  43750,  688205,  18352800, ...];
 n=6: [1, 36,  702, 10200, 133389, 1959552,  42189822, ...];
 n=7: [1, 49, 1274, 23863, 376320, 5810763, 108707676, ...];
 ...
where the terms along the main diagonal start as
 [1, 4, 54, 1280, 43750, 1959552, 108707676, ...]
which equals A000108(n-1)*n^n for n >= 1:
 [1, 1*2^2, 2*3^3, 5*4^4, 14*5^5, 42*6^6, 132*7^7, ...].
Compare the above table to the coefficients in 1/(1 - n*x)^n:
 n=1: [1,  1,    1,     1,      1,       1,         1, ...];
 n=2: [1,  4,   12,    32,     80,     192,       448, ...];
 n=3: [1,  9,   54,   270,   1215,    5103,     20412, ...];
 n=4: [1, 16,  160,  1280,   8960,   57344,    344064, ...];
 n=5: [1, 25,  375,  4375,  43750,  393750,   3281250, ...];
 n=6: [1, 36,  756, 12096, 163296, 1959552,  21555072, ...];
 n=7: [1, 49, 1372, 28812, 504210, 7764834, 108707676, ...];
 ...
to see that the main diagonals are equal.

Paul D. Hanna, Table of n, a(n) for n = 0..300

Cf. A263075, A263190, A171791.

Cf. A000108, A003714, A118113.

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

G.f. A(x) = Sum_{n>=0} a(n)*x^n satisfies the following formulas.
(1) [x^(n-1)] A(x)^(n^2) = n^n * binomial(2*n-1,n-1)/(2*n-1) for n >= 1.
(2) [x^(n-1)] A(x)^(n^2) = [x^(n-1)] 1/(1 - n*x)^n for n >= 1.

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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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A118113 Even Fibbinary numbers + 1; also 2*Fibbinary(n) + 1.

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A263075 G.f. satisfies: [x^(n-1)] A(x)^(n^2) = n^n * (n-1)! for n>=1.

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A294360 G.f. A(x) satisfies: [x^(n-1)] A(x)^(n^2) = (n^2)^(n-1) for n>=1.

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A371673 Expansion of g.f. A(x) satisfying [x^(n-1)] A(x)^(n^2) = A000108(n-1) * n^n for n >= 1, where A000108 is the Catalan numbers.

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