cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-5 of 5 results.

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

Original entry on oeis.org

1, 1, -4, 25, -194, 1603, -15264, 122316, -1897710, -8845133, -1169435932, -52853978047, -3193246498792, -205347570309000, -14534295599537024, -1115833257773950536, -92445637289048967654, -8219735646409095418617
Offset: 0

Views

Author

Paul D. Hanna, Jan 24 2010

Keywords

Comments

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

Examples

			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.

Links

Crossrefs

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

Programs

  • PARI
    {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), ", "))

Formula

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

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

Views

Author

Paul D. Hanna, Oct 11 2013

Keywords

Examples

			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; ...

Links

Crossrefs

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

Programs

  • PARI
    {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),", "))

Formula

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

A303562 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, 6, 50, 490, 5187, 59080, 675012, 8723880, 84841130, 2106192682, -26974249302, 2765793096248, -163142299607490, 11813146551718560, -906751607066476056, 75382006693375808940, -6718584345560312459292, 639573513055226901933760, -64760465046707144137421880, 6950351671309757070230871462
Offset: 0

Views

Author

Paul D. Hanna, Apr 27 2018

Keywords

Comments

a(n) is odd iff n = (4^k - 1)/3 for k >= 0 (conjecture).

Examples

			G.f. A(x) = 1 + x + 6*x^2 + 50*x^3 + 490*x^4 + 5187*x^5 + 59080*x^6 + 675012*x^7 + 8723880*x^8 + 84841130*x^9 + ...
such that the coefficient of x^(n-1) in 1/A(x)^(n^2+n-1) equals zero for n>2.
RELATED SERIES.
(x*A(x))' = 1 + 2*x + 18*x^2 + 200*x^3 + 2450*x^4 + 31122*x^5 + 413560*x^6 + 5400096*x^7 + 78514920*x^8 + ...
A'(x)/A(x) = 1 + 11*x + 133*x^2 + 1711*x^3 + 22386*x^4 + 304601*x^5 + 4019310*x^6 + 59971671*x^7 + 620401840*x^8 + ...
ILLUSTRATION OF DEFINITION.
The table of coefficients of x^k in 1/A(x)^(n^2+n-1) begins:
n=1: [1, -1, -5, -39, -371, -3842, -43425, -485860, ...];
n=2: [1, -5, -15, -105, -970, -9711, -110550, -1167485, ...];
n=3: [1, -11, 0, -44, -561, -5544, -74778, -601920, ...];
n=4: [1, -19, 76, 0, -95, 418, -27474, 277628, ...];
n=5: [1, -29, 261, -725, 0, 2871, -40716, 915501, ...];
n=6: [1, -41, 615, -4059, 10619, 0, -109347, 2014330, ...];
n=7: [1, -55, 1210, -13530, 80080, -225071, 0, 4884440, ...];
n=8: [1, -71, 2130, -35074, 343924, -2020731, 6422944, 0, ...]; ...
in which the main diagonal is all 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))' / A(x)^(n*(n+1)) begins:
n=1: [1, 0, 5, 78, 1113, 15368, 217125, 2915160, ...];
n=2: [1, -4, -9, -42, -194, 0, 22110, 466994, 10357803, ...];
n=3: [1, -10, 0, -32, -357, -3024, -33990, -218880, ...];
n=4: [1, -18, 68, 0, -75, 308, -18798, 175344, ...];
n=5: [1, -28, 243, -650, 0, 2376, -32292, 694518, ...];
n=6: [1, -40, 585, -3762, 9583, 0, -93345, 1670420, ...];
n=7: [1, -54, 1166, -12792, 74256, -204610, 0, 4262784, ...];
n=8: [1, -70, 2070, -33592, 324548, -1878426, 5880160, 0, ...]; ...
in which the main diagonal is all zeros after the initial terms, illustrating that 0 = [x^(n-1)] (x*A(x))' / A(x)^(n*(n+1)) for n>2.
The table of coefficients of x^k in (x*A(x)^(n+1))' / A(x)^(n*(n+1)) begins:
n=1: [1, 2, 22, 266, 3422, 44772, 609202, 8038620, ...];
n=2: [1, 0, 12, 176, 2457, 33288, 469690, 6150600, ...];
n=3: [1, -4, 0, 44, 854, 12672, 201160, 2446320, ...];
n=4: [1, -10, 10, 0, 70, 1222, 43320, 135920, ...];
n=5: [1, -18, 78, -50, 0, -408, 13950, -460224, ...];
n=6: [1, -28, 252, -784, 497, 0, 13258, -547944, ...];
n=7: [1, -40, 592, -3944, 11172, -8176, 0, -526608, ...];
n=8: [1, -54, 1170, -12936, 76194, -220374, 194424, 0, ...]; ...
in which the main diagonal is all zeros after the initial term, illustrating that 0 = [x^(n-1)] (x*A(x)^(n+1))' / A(x)^(n*(n+1)) for n>1.

Links

Crossrefs

Cf. A292877, A303563.

Programs

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

Formula

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

A294359 a(n) = [x^n] F(x)^(-(n+1)^2) such that F(x) = F(x^2) + x*F(x^4), where F(x) = Sum_{n>=0} x^A003714(n) and A003714 is the Fibbinary numbers.

Original entry on oeis.org

1, -4, 36, -544, 12000, -353016, 13024690, -578027008, 29965705056, -1776380879600, 118487748235604, -8781184406967264, 715759620936227036, -63634560244855290488, 6127715132571003255000, -635341671628285381320704, 70567080867797749860480968, -8358996420744136578157248864, 1051888164647093035820630830470, -140135781917815169726696222119200, 19704058040921706609228103696785954
Offset: 0

Views

Author

Paul D. Hanna, Nov 03 2017

Keywords

Comments

It is conjectured that all terms are even after the initial '1'.
Fibbinary numbers are integers whose binary representation contains no consecutive ones (see A003714 for definition); it is unexpected that the characteristic function F(x) of the Fibbinary numbers would have only even coefficients of x^n in the negative square powers F(x)^(-(n+1)^2), as described by this sequence.

Examples

			Given the characteristic function of the Fibbinary numbers (A003714):
F(x) = 1 + x + x^2 + x^4 + x^5 + x^8 + x^9 + x^10 + x^16 + x^17 + x^18 + x^20 + x^21 + x^32 + x^33 + x^34 + x^36 + x^37 + x^40 + x^41 + x^42 + x^64 + x^65 + x^66 + x^68 + x^69 + x^72 + x^73 + x^74 + x^80 +...+ x^A003714(n) +...
such that F(x) = F(x^2) + x*F(x^4),
then this sequence equals the coefficients of x^n in F(x)^(-(n+1)^2).
ILLUSTRATION OF TERMS.
The table of coefficients of x^k in F(x)^(-n^2) begins:
n=1: [1, -1, 0, 1, -2, 1, 2, -4, 2, 3, -8, 7, 4, -16, 16, 2, -30, ...];
n=2: [1, -4, 6, 0, -19, 40, -26, -56, 166, -160, -110, 560, -705, ...];
n=3: [1, -9, 36, -75, 36, 279, -942, 1278, 531, -5956, 11700, ...];
n=4: [1, -16, 120, -544, 1548, -2192, -2720, 23936, -63426, 67984, ...];
n=5: [1, -25, 300, -2275, 12000, -45005, 112450, -116350, -441375, ...];
n=6: [1, -36, 630, -7104, 57573, -353016, 1668774, -5996664, ...];
n=7: [1, -49, 1176, -18375, 209426, -1846859, 13024690, -74680760, ...];
n=8: [1, -64, 2016, -41600, 631216, -7491392, 72180992, -578027008, ...]; ...
in which the main diagonal forms this sequence.
RELATED SEQUENCES.
Terms (-1)^n * a(n)/(n+1) begin:
[1, 2, 12, 136, 2400, 58836, 1860670, 72253376, 3329522784, 177638087960, ...].
Sequence A294475(n) = (-1)^n * a(n)/(n+1)^2 and begins:
[1, 1, 4, 34, 480, 9806, 265810, 9031672, 369946976, 17763808796, ...].

Links

Crossrefs

Cf. A085357, A003714, A294475, A292877.

Programs

  • Mathematica
    terms = 21; selfibb = Select[Range[terms], BitAnd[#, 2*#] == 0&]; lenfibb = Length[selfibb]; fibb[0] = 0; fibb[n_] := selfibb[[n]]; F[x_] = Sum[x^fibb[n], {n, 0, lenfibb}]; a[n_] := SeriesCoefficient[F[x]^(-(n + 1)^2), {x, 0, n}]; Array[a, terms, 0] (* Jean-François Alcover, Nov 04 2017 *)

Formula

a(n) = (-1)^n * n^2 * A294475(n).

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

Views

Author

Paul D. Hanna, Apr 27 2018

Keywords

Comments

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

Examples

			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.

Links

Crossrefs

Cf. A303562, A230218, A292877.

Programs

  • PARI
    {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),", "))

Formula

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.
Showing 1-5 of 5 results.

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