A171791 -id:A171791 - cp's OEIS frontend

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

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

A233312 Terms of A114994 which are c-equivalent to "c-squares" (A020330).

Original entry on oeis.org

0, 3, 10, 15, 36, 43, 43, 63, 136, 147, 170, 175, 147, 175, 175, 255, 528, 547, 586, 591, 586, 683, 683, 703, 547, 591, 683, 703, 591, 703, 703, 1023, 2080, 2115, 2186, 2191, 2340, 2347, 2347, 2367, 2186, 2347, 2730, 2735, 2347, 2735, 2735, 2815, 2115, 2191

Offset: 0

Vladimir Shevelev, Dec 07 2013

About c-equivalent see in comment in A233249.

a(n) is even iff A171791(n+1) is odd - holds for at least the first 1028 terms. The reason, put very briefly, is that: a(n) is even if and only if n is the double of a "fibbinary number". Cf. A267508. [Jörgen Backelin, Jan 15 2016 added by Jeremy Gardiner, Jan 26 2016]

			c-square of 5 in binary is (10)(1)(10)(1)~(10)(10)(1)(1) which is 43 in decimal. So a(5)=43.

Peter J. C. Moses, Table of n, a(n) for n = 0..2499
Index entries for sequences related to binary expansion of n

Cf. A020330, A114994, A171791, A233249, A267508, A268032.

More terms from Peter J. C. Moses, Dec 07 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

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

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 531 terms. [See also A263190 and A171791.]

			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

A263190 G.f. satisfies: [x^(n-1)] A(x)^(n^2) = (n!)^2 for n>=1.

Original entry on oeis.org

1, 1, 0, 1, 46, 1723, 81104, 4793304, 349869074, 31080492631, 3318717525832, 421195540320465, 62871475566985208, 10927921934497456588, 2191068343727736627744, 502384409006686040020572, 130687814451798554601790746, 38294333521028379285810681487, 12557951067433973525611840784048, 4581888866092825667058378205370595

Offset: 0

Paul D. Hanna, Oct 12 2015

nonn

CONJECTURES.
(1) Limit a(n)/(n!)^2 = 1/exp(1).
(2) There are no negative terms.
(3) ODD TERMS: 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 531 terms. [See also A171791 and A263075.]
Conjectures hold to at least a(1000). - Sean A. Irvine, Oct 21 2015

			G.f.: A(x) = 1 + x + x^3 + 46*x^4 + 1723*x^5 + 81104*x^6 + 4793304*x^7 +...
The coefficients in A(x)^(n^2) begin:
n=1: [1, 1, 0, 1, 46, 1723, 81104, 4793304, 349869074, ...];
n=2: [1, 4, 6, 8, 197, 7456, 345654, 20167888, 1458010566, ...];
n=3: [1, 9, 36, 93, 612, 19197, 866208, 49440834, 3515499819, ...];
n=4: [1, 16, 120, 576, 2796, 44656, 1803872, 99433344, ...];
n=5: [1, 25, 300, 2325, 14400, 130705, 3606800, 183492150, ...];
n=6: [1, 36, 630, 7176, 61821, 518400, 8260086, 332807184, ...];
n=7: [1, 49, 1176, 18473, 216482, 2154775, 25401600, 655445812, ...];
n=8: [1, 64, 2016, 41728, 642352, 8045248, 95405312, 1625702400, ...];
n=9: [1, 81, 3240, 85401, 1673946, 26315199, 360707040, 5266837404, 131681894400, ...]; ...
where the terms along the main diagonal begin:
[1, 4, 36, 576, 14400, 518400, 25401600, 1625702400, ..., (n!)^2, ...].
LOCATION OF ODD TERMS.
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..265

Cf. A263075, A294360, A171791, A003714, A118113.

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

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

A268032 Run lengths in the parity of A233312.

Original entry on oeis.org

1, 1, 1, 1, 1, 3, 1, 1, 1, 5, 1, 1, 1, 1, 1, 11, 1, 1, 1, 1, 1, 3, 1, 1, 1, 21, 1, 1, 1, 1, 1, 3, 1, 1, 1, 5, 1, 1, 1, 1, 1, 43, 1, 1, 1, 1, 1, 3, 1, 1, 1, 5, 1, 1, 1, 1, 1, 11, 1, 1, 1, 1, 1, 3, 1, 1, 1, 85, 1, 1, 1, 1, 1, 3, 1, 1, 1, 5, 1, 1, 1, 1, 1, 11, 1, 1, 1, 1, 1, 3, 1, 1, 1, 21, 1, 1, 1, 1, 1, 3, 1, 1, 1, 5, 1, 1, 1, 1, 1, 171, 1

Offset: 1

Jeremy Gardiner, Jan 24 2016

nonn

Allowing for offset, also run lengths in the parity of A171791.
Records appear to be given by A001045 Jacobsthal numbers.
Run lengths of repeated 1's = 5,3,5,5,3,5,3,5,5,3,5, appears to be A194584.

			Parity of A233312 begins 0,1,0,1,0,1,1,1,0,1,0,1,1,1,1,1,0,1,0,1,0, ...

Jeremy Gardiner, Table of n, a(n) for n = 1..397

Cf. A001045, A171791, A194584, A268034.

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

cp's OEIS Frontend

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

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A233312 Terms of A114994 which are c-equivalent to "c-squares" (A020330).

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

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A263190 G.f. satisfies: [x^(n-1)] A(x)^(n^2) = (n!)^2 for n>=1.

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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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