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

A346648 a(n) = Sum_{k=0..n} binomial(n,k) * binomial(6*k,k) / (5*k + 1).

Original entry on oeis.org

1, 2, 9, 73, 751, 8587, 104425, 1323952, 17303503, 231455104, 3153167249, 43597546197, 610232050453, 8629733401556, 123114479858631, 1769728635257503, 25607523627970183, 372688563309335806, 5451995469296025115, 80122698147986922194, 1182341393088427774071
Offset: 0

Views

Author

Ilya Gutkovskiy, Jul 26 2021

Keywords

Comments

Binomial transform of A002295.

Links

Crossrefs

Cf. A002295, A007317, A188687, A226910, A346646, A346647, A346649, A346650.

Programs

  • Mathematica
    Table[Sum[Binomial[n, k] 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)^4 A[x]^6 + O[x]^(nmax + 1) // Normal, nmax + 1]; CoefficientList[A[x], x]
nmax = 20; CoefficientList[Series[Sum[(Binomial[6 k, k]/(5 k + 1)) x^k/(1 - x)^(k + 1), {k, 0, nmax}], {x, 0, nmax}], x]
Table[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, 20}]
  • PARI
    a(n) = sum(k=0, n, binomial(n,k)*binomial(6*k,k)/(5*k + 1)); \\ Michel Marcus, Jul 26 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) ~ 49781^(n + 3/2) / (3359232 * sqrt(3*Pi) * n^(3/2) * 5^(5*n + 3/2)). - Vaclav Kotesovec, Jul 30 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

A349293 G.f. A(x) satisfies A(x) = 1 / ((1 - x) * (1 - x * A(x)^7)).

Original entry on oeis.org

1, 2, 17, 249, 4345, 83285, 1694273, 35915349, 784691569, 17545398747, 399545961817, 9234298584921, 216053290499201, 5107287712887563, 121795876378121121, 2926604574330886897, 70788399943851406825, 1722188546498276868124, 42114624858397590035177
Offset: 0

Views

Author

Ilya Gutkovskiy, Nov 13 2021

Keywords

Comments

In general, for k>=1, Sum_{j=0..n} binomial(n + (k-1)*j,k*j) * binomial((k+1)*j,j) / (k*j+1) ~ sqrt(1 + (k-1)*r) / ((k+1)^(1/k) * sqrt(2*k*(k+1)*Pi*(1-r)) * n^(3/2) * r^(n + 1/k)), where r is the smallest real root of the equation (k+1)^(k+1) * r = k^k * (1-r)^k. - Vaclav Kotesovec, Nov 14 2021

Links

Crossrefs

Cf. A007317, A007556, A199475, A346650, A349289, A349290, A349291, A349292.

Programs

  • Mathematica
    nmax = 18; A[] = 0; Do[A[x] = 1/((1 - x) (1 - x A[x]^7)) + O[x]^(nmax + 1) // Normal, nmax + 1]; CoefficientList[A[x], x]
Table[Sum[Binomial[n + 6 k, 7 k] Binomial[8 k, k]/(7 k + 1), {k, 0, n}], {n, 0, 18}]
  • PARI
    a(n) = sum(k=0, n, binomial(n+6*k,7*k) * binomial(8*k,k) / (7*k+1)); \\ Michel Marcus, Nov 14 2021

Formula

a(n) = Sum_{k=0..n} binomial(n+6*k,7*k) * binomial(8*k,k) / (7*k+1).
a(n) ~ sqrt(1 + 6*r) / (2^(17/7) * sqrt(7*Pi*(1-r)) * n^(3/2) * r^(n + 1/7)), where r = 0.0375502499742240443056934699070050852345109331376051496159609551... is the real root of the equation 8^8 * r = 7^7 * (1-r)^7. - Vaclav Kotesovec, Nov 14 2021
a(n) = 1 + Sum_{x_1, x_2, ..., x_8>=0 and x_1+x_2+...+x_8=n-1} Product_{k=1..8} a(x_k). - Seiichi Manyama, Jul 11 2025

A346668 a(n) = Sum_{k=0..n} (-1)^(n-k) * binomial(n,k) * binomial(8*k,k) / (7*k + 1).

Original entry on oeis.org

1, 0, 7, 70, 917, 12922, 192591, 2984156, 47594289, 776184997, 12884436285, 216981375849, 3698021707457, 63663537870121, 1105474964523293, 19339098305850757, 340519405008643561, 6030158137055187758, 107328892461895007043, 1918980244360791943044, 34450128513971163342013
Offset: 0

Views

Author

Ilya Gutkovskiy, Jul 27 2021

Keywords

Comments

Inverse binomial transform of A007556.
In general, for m > 1, Sum_{k=0..n} (-1)^(n-k) * 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. A005043, A007556, A346628, A346650, A346664, A346665, A346666, A346667.

Programs

  • Mathematica
    Table[Sum[(-1)^(n - k) Binomial[n, k] Binomial[8 k, k]/(7 k + 1), {k, 0, n}], {n, 0, 20}]
nmax = 20; 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 = 20; CoefficientList[Series[Sum[(Binomial[8 k, k]/(7 k + 1)) x^k/(1 + x)^(k + 1), {k, 0, nmax}], {x, 0, nmax}], x]
Table[(-1)^n 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, 20}]
  • PARI
    a(n) = sum(k=0, n, (-1)^(n-k)*binomial(n,k)*binomial(8*k,k)/(7*k + 1)); \\ Michel Marcus, Jul 28 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) ~ 15953673^(n + 3/2) / (34359738368 * sqrt(Pi) * n^(3/2) * 7^(7*n + 3/2)). - Vaclav Kotesovec, Jul 30 2021

A349314 G.f. A(x) satisfies: A(x) = (1 + x * A(x)^8) / (1 - x).

Original entry on oeis.org

1, 2, 18, 274, 4930, 97346, 2039570, 44524818, 1001773058, 23065953794, 540886665618, 12872727013522, 310135678438978, 7549240857128258, 185381380643501970, 4586875745951650706, 114244031335228433922, 2862001783406012428802, 72067481493990612275474
Offset: 0

Views

Author

Ilya Gutkovskiy, Nov 14 2021

Keywords

Comments

In general, for k > 1, Sum_{j=0..n} binomial(n + (k-1)*j,k*j) * binomial(k*j,j) / ((k-1)*j+1) ~ (1-r)^(1/(k-1) - 1/2) * sqrt(1 + (k-1)*r) / (sqrt(2*Pi*(k-1)) * k^(1/(k-1) + 1/2) * n^(3/2) * r^(n + 1/(k-1))), where r is the smallest real root of the equation (k-1)^(k-1) * (1-r)^k = k^k * r. - Vaclav Kotesovec, Nov 15 2021

Links

Crossrefs

Cf. A006318, A007556, A346626, A346650, A349310, A349311, A349312, A349313.

Programs

  • Mathematica
    nmax = 18; A[] = 0; Do[A[x] = (1 + x A[x]^8)/(1 - x) + O[x]^(nmax + 1) // Normal, nmax + 1]; CoefficientList[A[x], x]
Table[Sum[Binomial[n + 7 k, 8 k] Binomial[8 k, k]/(7 k + 1), {k, 0, n}], {n, 0, 18}]

Formula

a(n) = Sum_{k=0..n} binomial(n+7*k,8*k) * binomial(8*k,k) / (7*k+1).
a(n) = F([(1+n)/7, (2+n)/7, (3+n)/7, (4+n)/7, (5+n)/7, (6+n)/7, 1+n/7, -n], [2/7, 3/7, 4/7, 5/7, 6/7, 1, 8/7], -1), where F is the generalized hypergeometric function. - Stefano Spezia, Nov 14 2021
a(n) ~ sqrt(1 + 7*r) / (2^(17/7) * sqrt(7*Pi) * (1-r)^(5/14) * n^(3/2) * r^(n + 1/7)), where r = 0.036466941615119756839260438459647497790132092200414533994... is the smallest real root of the equation 7^7 * (1-r)^8 = 8^8 * r. - Vaclav Kotesovec, Nov 15 2021

A349335 G.f. A(x) satisfies A(x) = 1 + x * A(x)^8 / (1 - x).

Original entry on oeis.org

1, 1, 9, 109, 1541, 23823, 390135, 6651051, 116798643, 2098313686, 38382509118, 712447023590, 13385500614902, 254065657922154, 4864482597112186, 93840443376075810, 1822169236520766546, 35586928273002974487, 698572561837366684479, 13775697096997873764647
Offset: 0

Views

Author

Ilya Gutkovskiy, Nov 15 2021

Keywords

Comments

In general, for m > 1, Sum_{k=0..n} binomial(n-1,k-1) * binomial(m*k,k) / ((m-1)*k+1) ~ (m-1)^(m/2 - 2) * (1 + m^m/(m-1)^(m-1))^(n + 1/2) / (sqrt(2*Pi) * m^((m-1)/2) * n^(3/2)). - Vaclav Kotesovec, Nov 15 2021

Links

Crossrefs

Cf. A002212, A307678, A349331, A349332, A349333, A349334.
Cf. A007556, A346650 (partial sums), A349364.

Programs

  • Maple
    a:= n-> coeff(series(RootOf(1+x*A^8/(1-x)-A, A), x, n+1), x, n):
seq(a(n), n=0..19);  # Alois P. Heinz, Nov 15 2021
  • Mathematica
    nmax = 19; A[] = 0; Do[A[x] = 1 + x A[x]^8/(1 - x) + O[x]^(nmax + 1) // Normal, nmax + 1]; CoefficientList[A[x], x]
Table[Sum[Binomial[n - 1, k - 1] Binomial[8 k, k]/(7 k + 1), {k, 0, n}], {n, 0, 19}]
  • 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)^8, 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(8*k,k) / (7*k+1).
a(n) ~ 17600759^(n + 1/2) / (2048 * sqrt(Pi) * n^(3/2) * 7^(7*n + 3/2)). - Vaclav Kotesovec, Nov 15 2021

A346672 a(n) = Sum_{k=0..n} binomial(8*k,k) / (7*k + 1).

Original entry on oeis.org

1, 2, 10, 102, 1342, 19620, 305004, 4943352, 82595376, 1412486081, 24602515801, 434935956337, 7783978950825, 140752989839105, 2567623696254905, 47195200645619009, 873239636055018809, 16251426606785706209, 304007720310330530081, 5713101394865420846381
Offset: 0

Views

Author

Ilya Gutkovskiy, Jul 28 2021

Keywords

Comments

Partial sums of A007556.
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 30 2021

Links

Crossrefs

Cf. A007556, A014137, A104859, A345367, A345368, A346065, A346650, A346671.

Programs

  • Mathematica
    Table[Sum[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)^7 A[x]^8 + O[x]^(nmax + 1) // Normal, nmax + 1]; CoefficientList[A[x], x]
  • PARI
    a(n) = sum(k=0, n, binomial(8*k, k)/(7*k+1)); \\ Michel Marcus, Jul 28 2021

Formula

G.f. A(x) satisfies: A(x) = 1 / (1 - x) + x * (1 - x)^7 * A(x)^8.
a(n) ~ 2^(24*n + 25) / (15953673 * sqrt(Pi) * n^(3/2) * 7^(7*n + 3/2)). - Vaclav Kotesovec, Jul 30 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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