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

A346646 a(n) = Sum_{k=0..n} binomial(n,k) * binomial(4*k,k) / (3*k + 1).

Original entry on oeis.org

1, 2, 7, 38, 257, 1935, 15505, 129519, 1115061, 9823160, 88121887, 802227794, 7392428009, 68819554003, 646276497617, 6114880542117, 58237420303109, 557850829527246, 5370956411708779, 51947475492561014, 504492516832543885, 4917564488572565160
Offset: 0

Views

Author

Ilya Gutkovskiy, Jul 26 2021

Keywords

Comments

Binomial transform of A002293.

Links

Crossrefs

Cf. A002293, A007317, A104859, A188687, A226974, A346647, A346648, A346649, A346650.

Programs

  • Maple
    A346646 := proc(n)
    hypergeom([-n,1/4,1/2,3/4],[2/3,1,4/3],-256/27) ;
    simplify(%) ;
end proc:
seq(A346646(n),n=0..40) ; # R. J. Mathar, Jan 10 2023
  • Mathematica
    Table[Sum[Binomial[n, k] Binomial[4 k, k]/(3 k + 1), {k, 0, n}], {n, 0, 21}]
nmax = 21; A[] = 0; Do[A[x] = 1/(1 - x) + x (1 - x)^2 A[x]^4 + O[x]^(nmax + 1) // Normal, nmax + 1]; CoefficientList[A[x], x]
nmax = 21; CoefficientList[Series[Sum[(Binomial[4 k, k]/(3 k + 1)) x^k/(1 - x)^(k + 1), {k, 0, nmax}], {x, 0, nmax}], x]
Table[HypergeometricPFQ[{1/4, 1/2, 3/4, -n}, {2/3, 1, 4/3}, -256/27], {n, 0, 21}]
  • PARI
    a(n) = sum(k=0, n, binomial(n,k)*binomial(4*k,k)/(3*k + 1)); \\ Michel Marcus, Jul 26 2021

Formula

G.f. A(x) satisfies: A(x) = 1 / (1 - x) + x * (1 - x)^2 * A(x)^4.
G.f.: Sum_{k>=0} ( binomial(4*k,k) / (3*k + 1) ) * x^k / (1 - x)^(k+1).
a(n) ~ 283^(n + 3/2) / (2048 * sqrt(2*Pi) * n^(3/2) * 3^(3*n + 3/2)). - Vaclav Kotesovec, Jul 30 2021
D-finite with recurrence 3*n*(3*n-1)*(3*n+1)*a(n) -2*(2*n-1) *(91*n^2 -91*n +24)*a(n-1) +6*(n-1) *(155*n^2 -310*n +167)*a(n-2) -438*(n-1) *(n-2)*(2*n-3) *a(n-3) +283*(n-1)*(n-2) *(n-3)*a(n-4)=0. - R. J. Mathar, Aug 17 2023

A346647 a(n) = Sum_{k=0..n} binomial(n,k) * binomial(5*k,k) / (4*k + 1).

Original entry on oeis.org

1, 2, 8, 54, 460, 4361, 43988, 462580, 5014252, 55624944, 628432101, 7205500484, 83632219892, 980710882430, 11601345881748, 138278231052451, 1659037424218780, 20020306637339944, 242835190201382648, 2958961154058610552, 36203518795424475661
Offset: 0

Views

Author

Ilya Gutkovskiy, Jul 26 2021

Keywords

Comments

Binomial transform of A002294.

Links

Crossrefs

Cf. A002294, A007317, A188687, A227035, A346646, A346648, A346649, A346650.

Programs

  • Maple
    A346647 := proc(n)
    hypergeom([-n,1/5,2/5,3/5,4/5],[1/2,3/4,1,5/4],-3125/256) ;
    simplify(%) ;
end proc:
seq(A346647(n),n=0..40) ; # R. J. Mathar, Jan 10 2023
  • Mathematica
    Table[Sum[Binomial[n, k] Binomial[5 k, k]/(4 k + 1), {k, 0, n}], {n, 0, 20}]
nmax = 20; A[] = 0; Do[A[x] = 1/(1 - x) + x (1 - x)^3 A[x]^5 + O[x]^(nmax + 1) // Normal, nmax + 1]; CoefficientList[A[x], x]
nmax = 20; CoefficientList[Series[Sum[(Binomial[5 k, k]/(4 k + 1)) x^k/(1 - x)^(k + 1), {k, 0, nmax}], {x, 0, nmax}], x]
Table[HypergeometricPFQ[{1/5, 2/5, 3/5, 4/5, -n}, {1/2, 3/4, 1, 5/4}, -3125/256], {n, 0, 20}]
  • PARI
    a(n) = sum(k=0, n, binomial(n,k)*binomial(5*k,k)/(4*k + 1)); \\ Michel Marcus, Jul 26 2021

Formula

G.f. A(x) satisfies: A(x) = 1 / (1 - x) + x * (1 - x)^3 * A(x)^5.
G.f.: Sum_{k>=0} ( binomial(5*k,k) / (4*k + 1) ) * x^k / (1 - x)^(k+1).
a(n) ~ 3381^(n + 3/2) / (78125 * sqrt(Pi) * n^(3/2) * 2^(8*n + 7/2)). - Vaclav Kotesovec, Jul 30 2021
D-finite with recurrence +8*n*(4*n+1) *(2*n-1)*(4*n-1)*a(n) +(-4405*n^4 +9322*n^3 -7655*n^2 +2978*n -480)*a(n-1) +12*(n-1) *(1255*n^3 -3829*n^2 +4204*n -1640) *a(n-2) -2*(n-1) *(n-2) *(10655*n^2 -32221*n +26076) *a(n-3) +4*(n-1) *(n-2) *(n-3)*(3445*n -6922) *a(n-4) -3381*(n-1)*(n-2) *(n-3)*(n-4)*a(n-5)=0. - R. J. Mathar, Aug 17 2023

A349312 G.f. A(x) satisfies: A(x) = (1 + x * A(x)^6) / (1 - x).

Original entry on oeis.org

1, 2, 14, 158, 2106, 30762, 476406, 7683926, 127692530, 2171184146, 37592376734, 660522703886, 11747865153962, 211093333172282, 3826315983647366, 69880933123237958, 1284661783610775010, 23753502514840942882, 441458929706855144494, 8242097867816771820926
Offset: 0

Views

Author

Ilya Gutkovskiy, Nov 14 2021

Keywords

Links

Crossrefs

Cf. A002295, A006318, A346626, A346648, A349310, A349311, A349313, A349314.

Programs

  • Mathematica
    nmax = 19; A[] = 0; Do[A[x] = (1 + x A[x]^6)/(1 - x) + O[x]^(nmax + 1) // Normal, nmax + 1]; CoefficientList[A[x], x]
Table[Sum[Binomial[n + 5 k, 6 k] Binomial[6 k, k]/(5 k + 1), {k, 0, n}], {n, 0, 19}]

Formula

a(n) = Sum_{k=0..n} binomial(n+5*k,6*k) * binomial(6*k,k) / (5*k+1).
a(n) = F([(1+n)/5, (2+n)/5, (3+n)/5, (4+n)/5, 1+n/5, -n], [2/5, 3/5, 4/5, 1, 6/5], -1), where F is the generalized hypergeometric function. - Stefano Spezia, Nov 14 2021
a(n) ~ sqrt(1 + 5*r) / (2^(6/5) * 3^(7/10) * sqrt(5*Pi) * (1-r)^(3/10) * n^(3/2) * r^(n + 1/5)), where r = 0.04941755525635041337247049893940451999923592381716... is the smallest real root of the equation 5^5 * (1-r)^6 = 6^6 * r. - Vaclav Kotesovec, Nov 15 2021

A349333 G.f. A(x) satisfies A(x) = 1 + x * A(x)^6 / (1 - x).

Original entry on oeis.org

1, 1, 7, 64, 678, 7836, 95838, 1219527, 15979551, 214151601, 2921712145, 40444378948, 566634504256, 8019501351103, 114484746457075, 1646614155398872, 23837794992712680, 347081039681365623, 5079306905986689309, 74670702678690897079, 1102218694940440851877
Offset: 0

Views

Author

Ilya Gutkovskiy, Nov 15 2021

Keywords

Links

Crossrefs

Cf. A002212, A307678, A349331, A349332, A349334, A349335.
Cf. A002295, A346648 (partial sums), A349362.

Programs

  • Maple
    a:= n-> coeff(series(RootOf(1+x*A^6/(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]^6/(1 - x) + O[x]^(nmax + 1) // Normal, nmax + 1]; CoefficientList[A[x], x]
Table[Sum[Binomial[n - 1, k - 1] Binomial[6 k, k]/(5 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)^6, 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(6*k,k) / (5*k+1).
a(n) ~ 49781^(n + 1/2) / (72 * sqrt(3*Pi) * n^(3/2) * 5^(5*n + 3/2)). - Vaclav Kotesovec, Nov 15 2021

A346649 a(n) = Sum_{k=0..n} binomial(n,k) * binomial(7*k,k) / (6*k + 1).

Original entry on oeis.org

1, 2, 10, 95, 1146, 15343, 218407, 3241316, 49588850, 776483636, 12383420161, 200444399493, 3284531747403, 54378741581471, 908238222519566, 15284835717461020, 258933935458506210, 4412025177612412048, 75564998345532498844, 1300158755391113561288
Offset: 0

Views

Author

Ilya Gutkovskiy, Jul 26 2021

Keywords

Comments

Binomial transform of A002296.

Links

Crossrefs

Cf. A002296, A007317, A188687, A346646, A346647, A346648, A346650.

Programs

  • Mathematica
    Table[Sum[Binomial[n, k] Binomial[7 k, k]/(6 k + 1), {k, 0, n}], {n, 0, 19}]
nmax = 19; A[] = 0; Do[A[x] = 1/(1 - x) + x (1 - x)^5 A[x]^7 + O[x]^(nmax + 1) // Normal, nmax + 1]; CoefficientList[A[x], x]
nmax = 19; CoefficientList[Series[Sum[(Binomial[7 k, k]/(6 k + 1)) x^k/(1 - x)^(k + 1), {k, 0, nmax}], {x, 0, nmax}], x]
Table[HypergeometricPFQ[{1/7, 2/7, 3/7, 4/7, 5/7, 6/7, -n}, {1/3, 1/2, 2/3, 5/6, 1, 7/6}, -823543/46656], {n, 0, 19}]
  • PARI
    a(n) = sum(k=0, n, binomial(n,k)*binomial(7*k,k)/(6*k + 1)); \\ Michel Marcus, Jul 26 2021

Formula

G.f. A(x) satisfies: A(x) = 1 / (1 - x) + x * (1 - x)^5 * A(x)^7.
G.f.: Sum_{k>=0} ( binomial(7*k,k) / (6*k + 1) ) * x^k / (1 - x)^(k+1).
a(n) ~ 870199^(n + 3/2) / (282475249 * sqrt(Pi) * n^(3/2) * 2^(6*n + 2) * 3^(6*n + 3/2)). - Vaclav Kotesovec, Jul 30 2021

A346650 a(n) = Sum_{k=0..n} binomial(n,k) * binomial(8*k,k) / (7*k + 1).

Original entry on oeis.org

1, 2, 11, 120, 1661, 25484, 415619, 7066670, 123865313, 2222178999, 40604688117, 753051711707, 14138552326609, 268204210248763, 5132686807360949, 98973130183436759, 1921142366704203305, 37508070639707177792, 736080632477073862271, 14511777729474947626918
Offset: 0

Views

Author

Ilya Gutkovskiy, Jul 26 2021

Keywords

Comments

Binomial transform of A007556.
In general, for m > 1, Sum_{k=0..n} binomial(n,k) * binomial(m*k,k) / ((m-1)*k + 1) ~ (m^m + (m-1)^(m-1))^(n + 3/2) / (sqrt(2*Pi) * m^((3*m-1)/2) * n^(3/2) * (m-1)^((m-1)*n + 3/2)). - Vaclav Kotesovec, Jul 30 2021

Links

Crossrefs

Cf. A007317, A007556, A188687, A346646, A346647, A346648, A346649.

Programs

  • Mathematica
    Table[Sum[Binomial[n, k] Binomial[8 k, k]/(7 k + 1), {k, 0, n}], {n, 0, 19}]
nmax = 19; A[] = 0; Do[A[x] = 1/(1 - x) + x (1 - x)^6 A[x]^8 + O[x]^(nmax + 1) // Normal, nmax + 1]; CoefficientList[A[x], x]
nmax = 19; CoefficientList[Series[Sum[(Binomial[8 k, k]/(7 k + 1)) x^k/(1 - x)^(k + 1), {k, 0, nmax}], {x, 0, nmax}], x]
Table[HypergeometricPFQ[{1/8, 1/4, 3/8, 1/2, 5/8, 3/4, 7/8, -n}, {2/7, 3/7, 4/7, 5/7, 6/7, 1, 8/7}, -16777216/823543], {n, 0, 19}]
  • PARI
    a(n) = sum(k=0, n, binomial(n,k)*binomial(8*k,k)/(7*k + 1)); \\ Michel Marcus, Jul 26 2021

Formula

G.f. A(x) satisfies: A(x) = 1 / (1 - x) + x * (1 - x)^6 * A(x)^8.
G.f.: Sum_{k>=0} ( binomial(8*k,k) / (7*k + 1) ) * x^k / (1 - x)^(k+1).
a(n) ~ 17600759^(n + 3/2) / (34359738368 * sqrt(Pi) * n^(3/2) * 7^(7*n + 3/2)). - Vaclav Kotesovec, Jul 30 2021

A349291 G.f. A(x) satisfies A(x) = 1 / ((1 - x) * (1 - x * A(x)^5)).

Original entry on oeis.org

1, 2, 13, 139, 1775, 24886, 370099, 5733304, 91518691, 1494815215, 24862931821, 419674102147, 7170713484877, 123783319369420, 2155542171446485, 37820343323942566, 667957770644685811, 11865421405897931581, 211856917750711562695, 3800040255017879663415
Offset: 0

Views

Author

Ilya Gutkovskiy, Nov 13 2021

Keywords

Links

Crossrefs

Cf. A002295, A007317, A199475, A346648, A349289, A349290, A349292, A349293.

Programs

  • Mathematica
    nmax = 19; A[] = 0; Do[A[x] = 1/((1 - x) (1 - x A[x]^5)) + O[x]^(nmax + 1) // Normal, nmax + 1]; CoefficientList[A[x], x]
Table[Sum[Binomial[n + 4 k, 5 k] Binomial[6 k, k]/(5 k + 1), {k, 0, n}], {n, 0, 19}]

Formula

a(n) = Sum_{k=0..n} binomial(n+4*k,5*k) * binomial(6*k,k) / (5*k+1).
a(n) ~ sqrt(1 + 4*r) / (2^(6/5) * 3^(7/10) * sqrt(5*Pi*(1-r)) * n^(3/2) * r^(n + 1/5)), where r = 0.051436794119208432185504972091697516647... is the real root of the equation 6^6 * r = 5^5 * (1-r)^5. - Vaclav Kotesovec, Nov 14 2021
a(n) = 1 + Sum_{x_1, x_2, ..., x_6>=0 and x_1+x_2+...+x_6=n-1} Product_{k=1..6} a(x_k). - Seiichi Manyama, Jul 10 2025

A346666 a(n) = Sum_{k=0..n} (-1)^(n-k) * binomial(n,k) * binomial(6*k,k) / (5*k + 1).

Original entry on oeis.org

1, 0, 5, 35, 335, 3405, 36601, 408630, 4693535, 55105970, 658390845, 7979041735, 97847884981, 1211946011450, 15139726594915, 190526268260405, 2413170608875655, 30738613968350640, 393519782671609951, 5060600804169151680, 65342131689498876095, 846781225288921612940
Offset: 0

Views

Author

Ilya Gutkovskiy, Jul 27 2021

Keywords

Comments

Inverse binomial transform of A002295.

Links

Crossrefs

Cf. A002295, A005043, A346628, A346648, A346664, A346665, A346667, A346668.

Programs

  • Mathematica
    Table[Sum[(-1)^(n - k) Binomial[n, k] Binomial[6 k, k]/(5 k + 1), {k, 0, n}], {n, 0, 21}]
nmax = 21; A[] = 0; Do[A[x] = 1/(1 + x) + x (1 + x)^4 A[x]^6 + O[x]^(nmax + 1) // Normal, nmax + 1]; CoefficientList[A[x], x]
nmax = 21; CoefficientList[Series[Sum[(Binomial[6 k, k]/(5 k + 1)) x^k/(1 + x)^(k + 1), {k, 0, nmax}], {x, 0, nmax}], x]
Table[(-1)^n HypergeometricPFQ[{1/6, 1/3, 1/2, 2/3, 5/6, -n}, {2/5, 3/5, 4/5, 1, 6/5}, 46656/3125], {n, 0, 21}]
  • PARI
    a(n) = sum(k=0, n, (-1)^(n-k)*binomial(n,k)*binomial(6*k,k)/(5*k + 1)); \\ Michel Marcus, Jul 28 2021

Formula

G.f. A(x) satisfies: A(x) = 1 / (1 + x) + x * (1 + x)^4 * A(x)^6.
G.f.: Sum_{k>=0} ( binomial(6*k,k) / (5*k + 1) ) * x^k / (1 + x)^(k+1).
a(n) ~ 43531^(n + 3/2) / (3359232 * sqrt(3*Pi) * n^(3/2) * 5^(5*n + 3/2)). - Vaclav Kotesovec, Jul 30 2021

A346065 a(n) = Sum_{k=0..n} binomial(6*k,k) / (5*k + 1).

Original entry on oeis.org

1, 2, 8, 59, 565, 6046, 68878, 818276, 10021910, 125629220, 1603943486, 20783993414, 272641113110, 3613484662965, 48313969712685, 650888627139801, 8826840286257595, 120398870546499685, 1650711840886884265, 22735860619151166130, 314441081323870331656
Offset: 0

Views

Author

Ilya Gutkovskiy, Jul 28 2021

Keywords

Comments

Partial sums of A002295.
In general, for m > 1, Sum_{k=0..n} binomial(m*k,k) / ((m-1)*k + 1) ~ m^(m*(n+1) + 1/2) / (sqrt(2*Pi) * (m^m - (m-1)^(m-1)) * n^(3/2) * (m-1)^((m-1)*n + 3/2)). - Vaclav Kotesovec, Jul 28 2021

Links

Crossrefs

Cf. A002295, A014137, A104859, A345367, A345368, A346648, A346671, A346672.

Programs

  • Mathematica
    Table[Sum[Binomial[6 k, k]/(5 k + 1), {k, 0, n}], {n, 0, 20}]
nmax = 20; A[] = 0; Do[A[x] = 1/(1 - x) + x (1 - x)^5 A[x]^6 + O[x]^(nmax + 1) // Normal, nmax + 1]; CoefficientList[A[x], x]
  • PARI
    a(n) = sum(k=0, n, binomial(6*k, k)/(5*k+1)); \\ Michel Marcus, Jul 28 2021

Formula

G.f. A(x) satisfies: A(x) = 1 / (1 - x) + x * (1 - x)^5 * A(x)^6.
a(n) ~ 2^(6*n + 6) * 3^(6*n + 13/2) / (43531 * sqrt(Pi) * n^(3/2) * 5^(5*n + 3/2)). - Vaclav Kotesovec, Jul 28 2021

A378327 a(n) = Sum_{k=0..n} binomial(n,k) * binomial(n*k,k) / ((n-1)*k + 1).

Original entry on oeis.org

1, 2, 5, 25, 257, 4361, 104425, 3241316, 123865313, 5628753361, 296671566941, 17798975341467, 1197924420178381, 89394126594968755, 7326377073291002147, 654215578855903951141, 63225054646397348577601, 6575059243843086616460321, 732138834180570978286488133
Offset: 0

Views

Author

Vaclav Kotesovec, Nov 23 2024

Keywords

Links

Crossrefs

Cf. A007317, A007556, A188687, A346646, A346647, A346648, A346649, A346650.
Cf. A378326.

Programs

  • Mathematica
    Table[Sum[Binomial[n, k] Binomial[n*k, k]/((n-1)*k + 1), {k, 0, n}], {n, 0, 20}]

Formula

a(n) ~ exp(n + exp(-1) - 1/2) * n^(n - 5/2) / sqrt(2*Pi).
Showing 1-10 of 11 results. Next

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