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 1-10 of 13 results. Next

A364742 G.f. satisfies A(x) = 1 / (1 - x*(1 + x*A(x))^3).

Original entry on oeis.org

1, 1, 4, 13, 50, 201, 841, 3627, 15993, 71803, 327082, 1508002, 7023446, 32995626, 156173668, 744029238, 3565030063, 17169013899, 83061503584, 403483653745, 1967217524551, 9623463731721, 47220968518786, 232354408276613, 1146254897566224, 5668118931395946
Offset: 0

Views

Author

Seiichi Manyama, Aug 05 2023

Keywords

Links

Crossrefs

Cf. A001006, A161634, A364743, A364744.

Programs

  • PARI
    a(n) = sum(k=0, n, binomial(n+1, k)*binomial(3*k, n-k))/(n+1);

Formula

a(n) = (1/(n+1)) * Sum_{k=0..n} binomial(n+1,k) * binomial(3*k,n-k).

A364743 G.f. satisfies A(x) = 1 / (1 - x*(1 + x*A(x))^4).

Original entry on oeis.org

1, 1, 5, 19, 85, 402, 1971, 9976, 51633, 272131, 1455486, 7879664, 43096967, 237777710, 1321792096, 7396125088, 41624735353, 235461758085, 1338049873395, 7634930866465, 43726638130854, 251273386911443, 1448362622788376, 8371936106228253
Offset: 0

Views

Author

Seiichi Manyama, Aug 05 2023

Keywords

Links

Crossrefs

Cf. A001006, A161634, A364742, A364744.

Programs

  • PARI
    a(n) = sum(k=0, n, binomial(n+1, k)*binomial(4*k, n-k))/(n+1);

Formula

a(n) = (1/(n+1)) * Sum_{k=0..n} binomial(n+1,k) * binomial(4*k,n-k).

A365118 G.f. satisfies A(x) = (1 + x / (1 - x*A(x)))^2.

Original entry on oeis.org

1, 2, 3, 8, 23, 72, 237, 808, 2830, 10118, 36779, 135510, 504935, 1899494, 7204238, 27517766, 105761937, 408715018, 1587169591, 6190357852, 24238696551, 95244997612, 375469654543, 1484519159122, 5885302251250, 23389997790804, 93172394487012
Offset: 0

Views

Author

Seiichi Manyama, Aug 22 2023

Keywords

Crossrefs

Cf. A001006, A365119.
Cf. A161634, A378801.

Programs

  • PARI
    a(n, s=1, t=2) = sum(k=0, n, binomial(t*(n-k+1), k)*binomial(n+(s-1)*k-1, n-k)/(n-k+1));

Formula

If g.f. satisfies A(x) = (1 + x/(1 - x*A(x))^s)^t, then a(n) = Sum_{k=0..n} binomial(t*(n-k+1),k) * binomial(n+(s-1)*k-1,n-k)/(n-k+1).
G.f.: A(x) = (1 + x*B(x))^2 where B(x) is the g.f. of A161634. - Seiichi Manyama, Dec 09 2024

A364744 G.f. satisfies A(x) = 1 / (1 - x*(1 + x*A(x))^5).

Original entry on oeis.org

1, 1, 6, 26, 131, 706, 3932, 22618, 133099, 797545, 4850296, 29859028, 185712831, 1165227025, 7366475715, 46877977451, 300049605259, 1930395961235, 12476394685445, 80968876247330, 527424073700966, 3447190219684125, 22599794010813360
Offset: 0

Views

Author

Seiichi Manyama, Aug 05 2023

Keywords

Links

Crossrefs

Cf. A001006, A161634, A364742, A364743.

Programs

  • PARI
    a(n) = sum(k=0, n, binomial(n+1, k)*binomial(5*k, n-k))/(n+1);

Formula

a(n) = (1/(n+1)) * Sum_{k=0..n} binomial(n+1,k) * binomial(5*k,n-k).

A367236 G.f. satisfies A(x) = 1 + x / (1 - x*A(x)^2)^2.

Original entry on oeis.org

1, 1, 2, 7, 26, 107, 462, 2074, 9572, 45147, 216638, 1054254, 5190710, 25810064, 129423512, 653740518, 3323270096, 16988894131, 87283137130, 450434292624, 2333851816654, 12136369892776, 63318984098996, 331347363084737, 1738713937163124, 9146850725274636
Offset: 0

Views

Author

Seiichi Manyama, Nov 11 2023

Keywords

Crossrefs

Cf. A001764, A367237, A367238.
Cf. A161634.

Programs

  • PARI
    a(n, s=2, t=0, u=2) = sum(k=0, n, binomial(t*k+u*(n-k)+1, k)*binomial(n+(s-1)*k-1, n-k)/(t*k+u*(n-k)+1));

Formula

If g.f. satisfies A(x) = 1 + x*A(x)^t / (1 - x*A(x)^u)^s, then a(n) = Sum_{k=0..n} binomial(t*k+u*(n-k)+1,k) * binomial(n+(s-1)*k-1,n-k) / (t*k+u*(n-k)+1).

A378801 G.f. A(x) satisfies A(x) = ( 1 + x/(1 - x*A(x)^(2/3)) )^3.

Original entry on oeis.org

1, 3, 6, 16, 48, 153, 511, 1761, 6219, 22383, 81804, 302766, 1132475, 4274166, 16256685, 62249167, 239772510, 928398831, 3611539758, 14107963848, 55318781982, 217652858539, 859027927911, 3400055112777, 13492710661658, 53673238384560, 213984657134418
Offset: 0

Views

Author

Seiichi Manyama, Dec 09 2024

Keywords

Crossrefs

Cf. A161634, A365118.
Cf. A005554, A378858.

Programs

  • PARI
    a(n, r=3, s=1, t=0, u=2) = r*sum(k=0, n, binomial(t*k+u*(n-k)+r, k)*binomial(n+(s-1)*k-1, n-k)/(t*k+u*(n-k)+r));

Formula

G.f. A(x) satisfies A(x) = 1 + x * (1 + A(x)^(1/3) + A(x)^(5/3)).
G.f.: A(x) = (1 + x*B(x))^3 where B(x) is the g.f. of A161634.
If g.f. satisfies A(x) = ( 1 + x*A(x)^(t/r) / (1 - x*A(x)^(u/r))^s )^r, then a(n) = r * Sum_{k=0..n} binomial(t*k+u*(n-k)+r,k) * binomial(n+(s-1)*k-1,n-k)/(t*k+u*(n-k)+r).

A364760 G.f. satisfies A(x) = 1 / (1 + x*(1 + x*A(x))^2).

Original entry on oeis.org

1, -1, -1, 4, 1, -21, 14, 111, -195, -529, 1837, 1792, -14772, 2300, 105431, -126697, -657427, 1650427, 3285795, -16211352, -8308737, 135770125, -79748628, -990431659, 1700106664, 6098396204, -20258923714, -27342511804, 193913175511, 12018867589
Offset: 0

Views

Author

Seiichi Manyama, Aug 05 2023

Keywords

Crossrefs

Cf. A007440, A364761, A364762, A364763.
Cf. A161634.

Programs

  • PARI
    a(n) = sum(k=0, n, (-1)^k*binomial(n+1, k)*binomial(2*k, n-k))/(n+1);

Formula

a(n) = (1/(n+1)) * Sum_{k=0..n} (-1)^k * binomial(n+1,k) * binomial(2*k,n-k).

A363560 Expansion of g.f. A(x) satisfying A(x)^3 = 1 + x*(A(x) + A(x)^2 + A(x)^9).

Original entry on oeis.org

1, 1, 3, 18, 126, 966, 7863, 66696, 583111, 5217513, 47547405, 439777242, 4117802109, 38956162023, 371795456373, 3575401032544, 34611064585803, 336998629754631, 3298200003722997, 32428037256038775, 320151289224740949, 3172536384239678856, 31544584654878015766
Offset: 0

Views

Author

Paul D. Hanna, Aug 12 2023

Keywords

Comments

Compare to: G(x)^3 = 1 + x*(G(x) + G(x)^2 + G(x)^3) holds when G(x) = 1/(1-x).
Conjecture: a(n) == 0 (mod 3) for n > 0 except when n == 1 (mod 7).

Examples

			G.f.: A(x) = 1 + x + 3*x^2 + 18*x^3 + 126*x^4 + 966*x^5 + 7863*x^6 + 66696*x^7 + 583111*x^8 + 5217513*x^9 + 47547405*x^10 + ...
such that
A(x)^3 = 1 + x*(A(x) + A(x)^2 + A(x)^9).
Also,
A(x) = 1 + x*(A(x) - A(x)^3 + A(x)^4 - A(x)^6 + A(x)^7).
RELATED TABLE.
The table of coefficients in A(x)^n begins:
n=1: [1, 1,  3,  18,  126,   966,   7863,   66696, ...];
n=2: [1, 2,  7,  42,  297,  2292,  18738,  159450, ...];
n=3: [1, 3, 12,  73,  522,  4059,  33354,  284886, ...];
n=4: [1, 4, 18, 112,  811,  6360,  52566,  450888, ...];
n=5: [1, 5, 25, 160, 1175,  9301,  77370,  666780, ...];
n=6: [1, 6, 33, 218, 1626, 13002, 108919,  943524, ...];
n=7: [1, 7, 42, 287, 2177, 17598, 148540, 1293937, ...];
n=8: [1, 8, 52, 368, 2842, 23240, 197752, 1732928, ...];
n=9: [1, 9, 63, 462, 3636, 30096, 258285, 2277756, ...];
...
from which one can verify the formulas involving powers of A(x).
RELATED SERIES.
Let G(x) = 1 + Series_Reversion( x/(1 + x*(1+x)^2 + x*(1+x)^5) )
where
G(x) = 1 + x + 2*x^2 + 11*x^3 + 61*x^4 + 380*x^5 + 2502*x^6 + 17163*x^7 + 121312*x^8 + 877370*x^9 + 6461765*x^10 + ...
then
A(x) = G(x*A(x)),
and so
A(x) = (1/x) * Series_Reversion( x/G(x) );
thus,
x*A(x) = (A(x) - 1) / (1 + (A(x) - 1)*(A(x)^2 + A(x)^5) )
which is equivalent to
A(x) = 1 + x*(A(x) - A(x)^3 + A(x)^4 - A(x)^6 + A(x)^7).
TERMS MODULO 3.
It appears that a(n) == 0 (mod 3) for n > 0 except when n == 1 (mod 7).
The residues of a(7*k + 1) modulo 3, for k >= 0, begin
a(7*k + 1) (mod 3) = [1, 1, 1, 1, 0, 2, 1, 0, 0, 1, 1, 2, 0, 0, 2, 2, 0, 0, 1, 1, 0, 0, 0, 2, 0, 0, 0, 1, 1, 1, 1, 0, 1, 2, 0, 0, 0, 0, 0, 0, 0, 2, 2, 0, 0, 2, 2, 0, 0, 0, 0, 0, 0, 0, 1, 1, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 2, 2, 0, ...].

Links

Crossrefs

Cf. A300048, A161634.

Programs

  • PARI
    {a(n) = my(A=1+x); for(i=1, n, A = (1 + x*(A + A^2 + A^9) +x*O(x^n))^(1/3) ); polcoeff(A, n)}
for(n=0, 30, print1(a(n), ", "))

Formula

G.f. A(x) = Sum_{n>=0} a(n)*x^n may be defined by the following formulas.
(1) A(x) = 1 + x*(A(x) - A(x)^3 + A(x)^4 - A(x)^6 + A(x)^7).
(2) A(x)^2 = 1 + x*(A(x) + A(x)^2 - A(x)^3 + A(x)^5 - A(x)^6 + A(x)^8) .
(3) A(x)^3 = 1 + x*(A(x) + A(x)^2 + A(x)^9).
(4) A(x)^4 = 1 + x*(A(x) + A(x)^2 + A(x)^4 - A(x)^6 + A(x)^7 + A(x)^10).
(5) A(x)^5 = 1 + x*(A(x) + A(x)^2 + A(x)^4 + A(x)^5 - A(x)^6 + A(x)^8 + A(x)^11).
(6) A(x)^6 = 1 + x*(A(x) + A(x)^2 + A(x)^4 + A(x)^5 + A(x)^9 + A(x)^12).
(7) A(x) = (1/x) * Series_Reversion( x/(1 + Series_Reversion( x/(1 + x*(1+x)^2 + x*(1+x)^5) ) ) ).

A363573 Expansion of g.f. A(x) satisfying A(x) = 1 + x*(A(x) - A(x)^3 + A(x)^5).

Original entry on oeis.org

1, 1, 3, 16, 99, 670, 4804, 35855, 275635, 2167577, 17354844, 140994899, 1159398760, 9631155422, 80703507043, 681333999628, 5789823864323, 49484286592503, 425092050147999, 3668385302806058, 31786451503719132, 276447315011186576, 2412336247105063011, 21114946136742383146
Offset: 0

Views

Author

Paul D. Hanna, Aug 14 2023

Keywords

Examples

			G.f.: A(x) = 1 + x + 3*x^2 + 16*x^3 + 99*x^4 + 670*x^5 + 4804*x^6 + 35855*x^7 + 275635*x^8 + 2167577*x^9 + 17354844*x^10 + ...
such that
A(x) = 1 + x*(A(x) - A(x)^3 + A(x)^5).
RELATED TABLE.
The table of coefficients in A(x)^n begins:
n=1: [1, 1,  3,  16,   99,   670,   4804,   35855,   275635, ...];
n=2: [1, 2,  7,  38,  239,  1634,  11798,   88506,   683045, ...];
n=3: [1, 3, 12,  67,  429,  2967,  21594,  162945,  1263183, ...];
n=4: [1, 4, 18, 104,  679,  4756,  34922,  265244,  2066591, ...];
n=5: [1, 5, 25, 150, 1000,  7101,  52645,  402725,  3155125, ...];
n=6: [1, 6, 33, 206, 1404, 10116,  75775,  584148,  4603911, ...];
n=7: [1, 7, 42, 273, 1904, 13930, 105490,  819918,  6503553, ...];
n=8: [1, 8, 52, 352, 2514, 18688, 143152, 1122312,  8962615, ...];
n=9: [1, 9, 63, 444, 3249, 24552, 190326, 1505727, 12110400, ...];
...
from which one can verify the formulas involving powers of A(x).
RELATED SERIES.
Let G(x) = 1 + Series_Reversion( x/(1 + x*(1+x)^2 + x*(1+x)^3) )
where
G(x) = 1 + x + 2*x^2 + 9*x^3 + 42*x^4 + 219*x^5 + 1202*x^6 + 6867*x^7 + 40378*x^8 + 242782*x^9 + 1485836*x^10 + ...
then
A(x) = G(x*A(x)),
and so
A(x) = (1/x) * Series_Reversion( x/G(x) );
thus,
x*A(x) = (A(x) - 1) / (1 + (A(x) - 1)*(A(x)^2 + A(x)^3) )
which is equivalent to
A(x) = 1 + x*(A(x) - A(x)^3 + A(x)^5).

Crossrefs

Cf. A219537, A271469, A300048, A161634, A363560.

Programs

  • Maple
    a:= n-> coeff(series(RootOf(1-A+x*(A-A^3+A^5), A), x, n+1), x, n):
seq(a(n), n=0..23);  # Alois P. Heinz, Aug 14 2023
  • PARI
    {a(n) = my(A=1+x); for(i=1, n, A = 1 + x*(A - A^3 + A^5) +x*O(x^n) ); polcoeff(A, n)}
for(n=0, 30, print1(a(n), ", "))

Formula

G.f. A(x) = Sum_{n>=0} a(n)*x^n may be defined by the following formulas.
(1) A(x) = 1 + x*(A(x) - A(x)^3 + A(x)^5).
(2) A(x)^2 = 1 + x*(A(x) + A(x)^2 - A(x)^3 - A(x)^4 + A(x)^5 + A(x)^6).
(3) A(x)^3 = 1 + x*(A(x) + A(x)^2 - A(x)^4 + A(x)^6 + A(x)^7).
(4) A(x)^4 = 1 + x*(A(x) + A(x)^2 + A(x)^7 + A(x)^8).
(5) A(x)^5 = 1 + x*(A(x) + A(x)^2 + A(x)^5 + A(x)^8 + A(x)^9).
(6) A(x)^6 = 1 + x*(A(x) + A(x)^2 + A(x)^5 + A(x)^6 + A(x)^9 + A(x)^10).
(7) A(x)^7 = 1 + x*(A(x) + A(x)^2 + A(x)^5 + A(x)^6 + A(x)^7 + A(x)^10 + A(x)^11).
(8) A(x) = (1/x) * Series_Reversion( x/(1 + Series_Reversion( x/(1 + x*(1+x)^2 + x*(1+x)^3) ) ) ).
(9) A(x) = 1 / A(-x*A(x)^5).

A365129 G.f. satisfies A(x) = (1 + x*A(x)*(1 + x*A(x))^2)^2.

Original entry on oeis.org

1, 2, 9, 44, 240, 1386, 8346, 51802, 329086, 2129330, 13984095, 92974510, 624568680, 4232731050, 28904102829, 198688337014, 1373763563150, 9547516671684, 66660156446189, 467342635522698, 3288691828900768, 23220922841177476, 164465227646878689
Offset: 0

Views

Author

Seiichi Manyama, Aug 23 2023

Keywords

Crossrefs

Cf. A161634, A365130.
Cf. A000108, A365133.

Programs

  • PARI
    a(n, s=2, t=2) = sum(k=0, n, binomial(t*(n+1), k)*binomial(s*k, n-k))/(n+1);

Formula

If g.f. satisfies A(x) = (1 + x*A(x)*(1 + x*A(x))^s)^t, then a(n) = (1/(n+1)) * Sum_{k=0..n} binomial(t*(n+1),k) * binomial(s*k,n-k).
Showing 1-10 of 13 results. Next

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