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 11 results. Next

A349332 G.f. A(x) satisfies A(x) = 1 + x * A(x)^5 / (1 - x).

Original entry on oeis.org

1, 1, 6, 46, 406, 3901, 39627, 418592, 4551672, 50610692, 572807157, 6577068383, 76426719408, 897078662538, 10620634999318, 126676885170703, 1520759193166329, 18361269213121164, 222814883564042704, 2716125963857227904, 33244557641365865109
Offset: 0

Views

Author

Ilya Gutkovskiy, Nov 15 2021

Keywords

Links

Crossrefs

Cf. A002212, A307678, A349331, A349333, A349334, A349335.
Cf. A002294, A346647 (partial sums), A349361, A364748.

Programs

  • Maple
    a:= n-> coeff(series(RootOf(1+x*A^5/(1-x)-A, A), x, n+1), x, n):
seq(a(n), n=0..20);  # Alois P. Heinz, Nov 15 2021
  • Mathematica
    nmax = 20; A[] = 0; Do[A[x] = 1 + x A[x]^5/(1 - x) + O[x]^(nmax + 1) // Normal, nmax + 1]; CoefficientList[A[x], x]
Table[Sum[Binomial[n - 1, k - 1] Binomial[5 k, k]/(4 k + 1), {k, 0, n}], {n, 0, 20}]
  • PARI
    {a(n) = my(A=[1]); for(m=1, n, A=concat(A, 0);
A[#A] = 1 + sum(k=1, m-1, (polcoeff(Ser(A)^5, k)) )); A[n+1]}
for(n=0, 30, print1(a(n), ", ")) \\ Vaclav Kotesovec, Nov 23 2024, after Paul D. Hanna

Formula

a(n) = Sum_{k=0..n} binomial(n-1,k-1) * binomial(5*k,k) / (4*k+1).
a(n) ~ 3381^(n + 1/2) / (25 * sqrt(Pi) * n^(3/2) * 2^(8*n + 7/2)). - Vaclav Kotesovec, Nov 15 2021
Recurrence: 8*n*(2*n - 1)*(4*n - 1)*(4*n + 1)*a(n) = (4405*n^4 - 10346*n^3 + 9575*n^2 - 4354*n + 840)*a(n-1) - 12*(n-2)*(1255*n^3 - 3957*n^2 + 4492*n - 1820)*a(n-2) + 2*(n-3)*(n-2)*(10655*n^2 - 32733*n + 26908)*a(n-3) - 4*(n-4)*(n-3)*(n-2)*(3445*n - 6986)*a(n-4) + 3381*(n-5)*(n-4)*(n-3)*(n-2)*a(n-5). - Vaclav Kotesovec, Nov 17 2021

A364747 G.f. A(x) satisfies A(x) = 1 + x*A(x)^4 / (1 - x*A(x)).

Original entry on oeis.org

1, 1, 5, 32, 234, 1854, 15490, 134380, 1198944, 10931761, 101412677, 954155059, 9083120975, 87326765375, 846709605539, 8269910074087, 81291388929027, 803592049667495, 7983612883739843, 79671910265120574, 798283229227457304, 8027625597750959053
Offset: 0

Views

Author

Seiichi Manyama, Aug 05 2023

Keywords

Links

Crossrefs

Cf. A000108, A001003, A108447, A364748, A378691, A378692.
Cf. A002294, A243659, A364765, A364792.

Programs

  • PARI
    a(n) = if(n==0, 1, sum(k=0, n-1, binomial(n, k)*binomial(4*n-3*k, n-1-k))/n);
  • PARI
    a(n, r=1, s=1, t=4, u=1) = 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)); \\ Seiichi Manyama, Dec 05 2024

Formula

a(n) = (1/n) * Sum_{k=0..n-1} binomial(n,k) * binomial(4*n-3*k,n-1-k) for n > 0.
From Seiichi Manyama, Dec 05 2024: (Start)
G.f. A(x) satisfies A(x) = 1/(1 - x*A(x)^3/(1 - x*A(x))).
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). (End)

A364759 G.f. satisfies A(x) = 1 + x*A(x)^5 / (1 + x*A(x)).

Original entry on oeis.org

1, 1, 4, 25, 182, 1447, 12175, 106575, 960579, 8854622, 83089537, 791063172, 7622317663, 74191096721, 728389554533, 7204640725610, 71727367291455, 718195853746770, 7227785937663908, 73069500402699226, 741712341691454837, 7556704348506425398
Offset: 0

Views

Author

Seiichi Manyama, Aug 05 2023

Keywords

Crossrefs

Cf. A090192, A106228, A364758.
Cf. A364748.

Programs

  • PARI
    a(n) = if(n==0, 1, sum(k=0, n-1, (-1)^k*binomial(n, k)*binomial(5*n-4*k, n-1-k))/n);

Formula

a(n) = (1/n) * Sum_{k=0..n-1} (-1)^k * binomial(n,k) * binomial(5*n-4*k,n-1-k) for n > 0.

A365192 G.f. satisfies A(x) = 1 + x*A(x)^5 / (1 - x*A(x)^2).

Original entry on oeis.org

1, 1, 6, 48, 443, 4445, 47107, 518835, 5880223, 68130860, 803369481, 9609294542, 116310009888, 1421951861817, 17533301767624, 217796367181117, 2722942699583650, 34236790400004432, 432649744252128084, 5492060945760586212, 69998993052214823013
Offset: 0

Views

Author

Seiichi Manyama, Aug 25 2023

Keywords

Crossrefs

Cf. A002295, A243667, A349332, A364748, A365193, A365194.
Cf. A365186.

Programs

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

Formula

a(n) = Sum_{k=0..n} binomial(2*n+3*k+1,k) * binomial(n-1,n-k)/(2*n+3*k+1).

A378692 G.f. A(x) satisfies A(x) = 1 + x*A(x)^7/(1 - x*A(x)).

Original entry on oeis.org

1, 1, 8, 86, 1075, 14667, 211799, 3182454, 49243854, 779379652, 12558073022, 205312307834, 3397359326116, 56790504859929, 957574385205771, 16267419813629731, 278162968238908681, 4783813617177604232, 82691541747420586716, 1435895455224032519430, 25035634270828781060188
Offset: 0

Views

Author

Seiichi Manyama, Dec 04 2024

Keywords

Crossrefs

Cf. A000108, A001003, A108447, A364747, A364748, A378691.
Cf. A378686, A378690, A378693.
Cf. A378685, A378688, A378694.

Programs

  • PARI
    a(n, r=1, s=1, t=7, u=1) = 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/(1 - x*A(x)^6/(1 - x*A(x))).
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).

A365185 G.f. satisfies A(x) = 1 + x*A(x)^5*(1 + x*A(x)).

Original entry on oeis.org

1, 1, 6, 46, 411, 3996, 41062, 438662, 4823133, 54221518, 620404859, 7201317005, 84590041441, 1003656037278, 12010861830069, 144804336388912, 1757106190680819, 21443109365898743, 263009775111233392, 3240530659303505547, 40088688455992604594
Offset: 0

Views

Author

Seiichi Manyama, Aug 25 2023

Keywords

Crossrefs

Cf. A002295, A365184, A365186, A365187, A365188, A365189.
Cf. A364748.

Programs

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

Formula

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

A365193 G.f. satisfies A(x) = 1 + x*A(x)^5 / (1 - x*A(x)^3).

Original entry on oeis.org

1, 1, 6, 49, 463, 4760, 51702, 583712, 6781774, 80555066, 973813974, 11941861079, 148191437719, 1857464450449, 23481830726334, 299056887494427, 3833349330581255, 49416395972195630, 640256115370243620, 8332835556325119938, 108890550249605779116
Offset: 0

Views

Author

Seiichi Manyama, Aug 25 2023

Keywords

Crossrefs

Cf. A002295, A243667, A349332, A364748, A365192, A365194.
Cf. A365187.

Programs

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

Formula

a(n) = Sum_{k=0..n} binomial(3*n+2*k+1,k) * binomial(n-1,n-k)/(3*n+2*k+1).

A365194 G.f. satisfies A(x) = 1 + x*A(x)^5 / (1 - x*A(x)^6).

Original entry on oeis.org

1, 1, 6, 52, 529, 5889, 69462, 853013, 10791018, 139659604, 1840435530, 24611295075, 333132371248, 4555465710569, 62839303262352, 873363902976309, 12218178082489873, 171918448407833112, 2431415226089290680, 34544425914499450493, 492807213597429920649
Offset: 0

Views

Author

Seiichi Manyama, Aug 25 2023

Keywords

Crossrefs

Cf. A002295, A243667, A349332, A364748, A365192, A365193.
Cf. A219537, A271469, A364765.

Programs

  • PARI
    a(n) = sum(k=0, n, binomial(6*n-k+1, k)*binomial(n-1, n-k)/(6*n-k+1));

Formula

a(n) = Sum_{k=0..n} binomial(6*n-k+1,k) * binomial(n-1,n-k)/(6*n-k+1).
a(n) = Sum_{k=0..n} (-1)^(n-k) * binomial(5*n+2*k+1,k) * binomial(n-1,n-k)/(5*n+2*k+1).
a(n) = (1/n) * Sum_{k=0..floor((n-1)/2)} binomial(n,k) * binomial(6*n-k,n-1-2*k) for n > 0. - Seiichi Manyama, Dec 26 2024

A371890 G.f. A(x) satisfies A(x) = 1 - x/A(x)^3 * (1 - A(x) - A(x)^4).

Original entry on oeis.org

1, 1, 2, 1, -4, 0, 37, 16, -313, -214, 3005, 2943, -30391, -39432, 318606, 522863, -3418205, -6889626, 37219105, 90415336, -408758113, -1183054415, 4505089166, 15442590040, -49599878555, -201138280510, 542949788652, 2614332298108, -5877502079248
Offset: 0

Views

Author

Seiichi Manyama, Apr 11 2024

Keywords

Crossrefs

Cf. A349331, A364748, A367724, A371891.

Programs

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

Formula

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

A378691 G.f. A(x) satisfies A(x) = 1 + x*A(x)^6/(1 - x*A(x)).

Original entry on oeis.org

1, 1, 7, 65, 699, 8192, 101538, 1309007, 17373825, 235820907, 3258327727, 45676003435, 648019627185, 9286982935406, 134247731827970, 1955128344950960, 28659409029300490, 422517375650417841, 6260750426764454787, 93191618760715641120, 1392823412892172416996
Offset: 0

Views

Author

Seiichi Manyama, Dec 04 2024

Keywords

Crossrefs

Cf. A000108, A001003, A108447, A364747, A364748, A378692.

Programs

  • PARI
    a(n, r=1, s=1, t=6, u=1) = 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/(1 - x*A(x)^5/(1 - x*A(x))).
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).
Showing 1-10 of 11 results. Next

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

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