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.

A206178 a(n) = Sum_{k=0..n} binomial(n,k)^3 * 2^k.

Original entry on oeis.org

1, 3, 21, 171, 1521, 14283, 138909, 1385163, 14072193, 145039923, 1512191781, 15914734443, 168802010001, 1802247516891, 19350710547021, 208783189719531, 2262263134211073, 24604815145831011, 268499713118585781, 2938736789722114731, 32250788066104022961
Offset: 0

Views

Author

Paul D. Hanna, Feb 04 2012

Keywords

Comments

Ignoring initial term, equals the logarithmic derivative of A206177.
Compare to Franel numbers: A000172(n) = Sum_{k=0..n} binomial(n,k)^3.
Diagonal of rational functions 1/(1 - x*y + y*z + 2*x*z - 3*x*y*z), 1/(1 + y + z + x*y + y*z + 2*x*z + 3*x*y*z), 1/(1 - x + 2*z + x*y - y*z - 2*x*z + 3*x*y*z), 1/(1 - x - y - z + x*y + y*z + x*z - 3*x*y*z), 1/(1 - x + y + 2*z - x*y + 2*y*z - 2*x*z - 3*x*y*z). - Gheorghe Coserea, Jul 03 2018

Examples

			L.g.f.: L(x) = 3*x + 21*x^2/2 + 171*x^3/3 + 1521*x^4/4 + 14283*x^5/5 +...
Exponentiation equals the g.f. of A206177:
exp(L(x)) = 1 + 3*x + 15*x^2 + 93*x^3 + 657*x^4 + 5067*x^5 + 41579*x^6 +...

Links

Crossrefs

Cf. A206177, A000172, A206180, A216483, A216636.
Related to diagonal of rational functions: A268545-A268555.

Programs

  • Mathematica
    Flatten[{1,RecurrenceTable[{(n+3)^2*(3*n+4)*a[n+3]-3*(9*n^3+57*n^2+116*n+74)*a[n+2]-3*(27*n^3+144*n^2+252*n+145)*a[n+1]-27*(3*n+7)*(n+1)^2*a[n]==0, a[1]==3, a[2]==21, a[3]==171},a,{n,1,20}]}] (* Vaclav Kotesovec, Sep 11 2012 *)
Table[HypergeometricPFQ[{-n, -n, -n}, {1, 1}, -2], {n, 0, 20}] (* Jean-François Alcover, Oct 25 2019 *)
  • PARI
    {a(n)=sum(k=0,n,binomial(n,k)^3*2^k)}
  • Sage
    A206178 = lambda n: hypergeometric([-n,-n,-n], [1,1], -2)
[Integer(A206178(n).n(100)) for n in (0..20)] # Peter Luschny, Sep 23 2014

Formula

a(2*3^n) == 3 (mod 9) for n>=0; a(n) == 0 (mod 9) if n/2 > 1 is not a power of 3.
Recurrence: (n+3)^2*(3*n+4)*a(n+3) - 3*(9*n^3+57*n^2+116*n+74)*a(n+2) - 3*(27*n^3+144*n^2+252*n+145)*a(n+1) - 27*(3*n+7)*(n+1)^2*a(n) = 0. - Vaclav Kotesovec, Sep 11 2012
a(n) ~ (1 + 2^(1/3))^(3*n + 2) / (2^(4/3)*sqrt(3)*Pi*n). - Vaclav Kotesovec, Sep 19 2012, simplified Apr 24 2025
G.f.: hypergeom([1/3, 2/3],[1],54*x^2/(1-3*x)^3)/(1-3*x). - Mark van Hoeij, May 02 2013
a(n) = hypergeom([-n,-n,-n],[1,1], -2). - Peter Luschny, Sep 23 2014
G.f. y=A(x) satisfies: 0 = x*(3*x + 2)*(27*x^3 + 27*x^2 + 9*x - 1)*y'' + (243*x^4 + 378*x^3 + 189*x^2 + 36*x - 2)*y' + 3*(x + 1)*(27*x^2 + 12*x + 2)*y. - Gheorghe Coserea, Jul 01 2018

Extensions

Minor edits by Vaclav Kotesovec, Mar 31 2014

A206180 a(n) = Sum_{k=0..n} binomial(n,k)^3 * 3^k.

Original entry on oeis.org

1, 4, 34, 352, 3946, 46744, 573616, 7217536, 92527738, 1203467464, 15834369244, 210304283776, 2815055712496, 37930536447808, 513972867056704, 6998587355233792, 95704396144575898, 1313665229153722408, 18091969874675059204, 249908773119244105792
Offset: 0

Views

Author

Paul D. Hanna, Feb 04 2012

Keywords

Comments

Ignoring initial term, equals the logarithmic derivative of A206179.
Compare to Franel numbers: A000172(n) = Sum_{k=0..n} binomial(n,k)^3.
Diagonal of rational function 1/(1 + y + z + x*y + y*z + 3*x*z + 4*x*y*z). - Gheorghe Coserea, Jul 01 2018
Diagonal of rational function 1 / ((1-x)*(1-y)*(1-z) - 3*x*y*z). - Seiichi Manyama, Jul 11 2020

Examples

			L.g.f.: L(x) = 4*x + 34*x^2/2 + 352*x^3/3 + 3946*x^4/4 + 46744*x^5/5 +...
Exponentiation equals the g.f. of A206179:
exp(L(x)) = 1 + 4*x + 25*x^2 + 196*x^3 + 1747*x^4 + 16996*x^5 + 175936*x^6 +...

Links

Crossrefs

Cf. A206179, A000172, A206178, A216483, A216636.

Programs

  • Mathematica
    Flatten[{1,RecurrenceTable[{(n+3)^2*(3*n+4)*a[n+3]-4*(9*n^3+57*n^2+116*n+74)*a[n+2]-(99*n^3+528*n^2+929*n+540)*a[n+1]-64*(3*n+7)*(n+1)^2*a[n]==0,a[1]==4,a[2]==34,a[3]==352},a,{n,1,20}]}] (* Vaclav Kotesovec, Sep 11 2012 *)
Table[HypergeometricPFQ[{-n, -n, -n}, {1, 1}, -3] , {n, 0, 20}] (* Jean-François Alcover, Oct 25 2019 *)
  • PARI
    {a(n)=sum(k=0,n,binomial(n,k)^3*3^k)}
for(n=0,41,print1(a(n),", "))
  • Sage
    A206180 = lambda n: hypergeometric([-n,-n,-n],[1,1], -3)
[Integer(A206180(n).n(100)) for n in (0..19)] # Peter Luschny, Sep 23 2014

Formula

Recurrence: (n+3)^2*(3*n+4)*a(n+3) - 4*(9*n^3+57*n^2+116*n+74)*a(n+2) - (99*n^3+528*n^2+929*n+540)*a(n+1) - 64*(3*n+7)*(n+1)^2*a(n) = 0. - Vaclav Kotesovec, Sep 11 2012
a(n) ~ (1 + 3^(1/3))^(3*n + 2) / (2*3^(5/6)*Pi*n). - Vaclav Kotesovec, Sep 19 2012, simplified Apr 24 2025
G.f.: hypergeom([1/3, 2/3],[1],81*x^2/(1-4*x)^3)/(1-4*x). - Mark van Hoeij, May 02 2013
a(n) = hypergeometric([-n,-n,-n],[1,1], -3). - Peter Luschny, Sep 23 2014
G.f. y=A(x) satisfies: 0 = x*(2*x + 1)*(64*x^3 + 33*x^2 + 12*x - 1)*y'' + (384*x^4 + 388*x^3 + 123*x^2 + 24*x - 1)*y' + (128*x^3 + 132*x^2 + 24*x + 4)*y. - Gheorghe Coserea, Jul 01 2018

Extensions

Minor edits by Vaclav Kotesovec, Mar 31 2014

A216483 a(n) = Sum_{k=0..n} binomial(n,k)^3 * 4^k.

Original entry on oeis.org

1, 5, 49, 605, 8065, 113525, 1656145, 24774125, 377601025, 5839329125, 91349718769, 1442580779645, 22959923825281, 367847984671445, 5926784048373265, 95960317086368525, 1560335109283897345, 25466972987548413125, 417048643127042376625, 6850021673230814868125
Offset: 0

Views

Author

Vaclav Kotesovec, Sep 11 2012

Keywords

Comments

Diagonal of rational function 1/(1 + y + z + x*y + y*z + 4*x*z + 5*x*y*z). - Gheorghe Coserea, Jul 01 2018
Diagonal of rational function 1 / ((1-x)*(1-y)*(1-z) - 4*x*y*z). - Seiichi Manyama, Jul 11 2020

Links

Crossrefs

Cf. A206178, A206180, A216636.

Programs

  • Mathematica
    Table[Sum[Binomial[n,k]^3*4^k,{k,0,n}],{n,0,20}]
  • PARI
    a(n) = sum(k=0, n, binomial(n,k)^3 * 4^k); \\ Gheorghe Coserea, Jul 01 2018
  • Sage
    A216483 = lambda n: hypergeometric([-n,-n,-n], [1,1], -4)
[Integer(A216483(n).n(100)) for n in (0..19)] # Peter Luschny, Sep 23 2014

Formula

Recurrence: (n+3)^2*(3*n+4)*a(n+3) = 5*(9*n^3+57*n^2+116*n+74)*a(n+2) + (99*n^3+528*n^2+938*n+555)*a(n+1) + 125*(3*n+7)*(n+1)^2*a(n).
a(n) ~ (1 + 2^(2/3))^(3*n+2) / (2^(5/3)*sqrt(3)*Pi*n). - Vaclav Kotesovec, Sep 19 2012, simplified Apr 24 2025
G.f.: hypergeom([1/3, 2/3],[1],108*x^2/(1-5*x)^3)/(1-5*x). - Mark van Hoeij, May 02 2013
a(n) = hypergeom([-n,-n,-n],[1,1],-4). - Peter Luschny, Sep 23 2014
G.f. y=A(x) satisfies: 0 = x*(5*x + 2)*(125*x^3 + 33*x^2 + 15*x - 1)*y'' + (1875*x^4 + 1330*x^3 + 273*x^2 + 60*x - 2)*y' + (625*x^3 + 495*x^2 + 42*x + 10)*y. - Gheorghe Coserea, Jul 01 2018

Extensions

Minor edits by Vaclav Kotesovec, Mar 31 2014

A216698 a(n) = Sum_{k=0..n} binomial(n,k)^3 * 6^k.

Original entry on oeis.org

1, 7, 85, 1351, 23281, 422527, 7951069, 153458935, 3018043777, 60225528727, 1215821974885, 24777776573095, 508935634491025, 10522995625652335, 218814097786515085, 4572338217781407031, 95953172529722919937, 2021236451413828339495, 42719661851354642952181
Offset: 0

Views

Author

Vaclav Kotesovec, Sep 15 2012

Keywords

Links

Crossrefs

Cf. A000172 (x=1), A206178 (x=2), A206180 (x=3), A216483 (x=4), A216636 (x=5), A216696.

Programs

  • Mathematica
    Table[Sum[Binomial[n, k]^3*6^k, {k, 0, n}], {n, 0, 25}]
  • Sage
    A216698 = lambda n: hypergeometric([-n,-n,-n], [1,1], -6)
[Integer(A216698(n).n(100)) for n in (0..18)] # Peter Luschny, Sep 23 2014

Formula

General recurrecnce for Sum_{k=0..n} binomial(n,k)^3*x^k (this is case x=6): (n+3)^2*(3*n+4)*a(n+3) -(9*n^3+57*n^2+116*n+74)*(x+1)*a(n+2) +(3*n+5)*(3*n^2*(x^2-7*x+1)+11*n*(x^2-7*x+1)+9*x^2-66*x+9)*a(n+1) -(n+1)^2*(3*n+7)*(x+1)^3*a(n) = 0.
a(n) ~ (1+6^(1/3))^2/(2*2^(1/3)*3^(5/6)*Pi) * (1+6^(1/3))^(3*n)/n. - Vaclav Kotesovec, Sep 19 2012
G.f.: hypergeom([1/3, 2/3],[1],6*27*x^2/(1-7*x)^3)/(1-7*x). - Mark van Hoeij, May 02 2013
a(n) = hypergeom([-n,-n,-n],[1,1], -6). - Peter Luschny, Sep 23 2014

Extensions

Minor edits by Vaclav Kotesovec, Mar 31 2014

A336163 Square array T(n,k), n >= 0, k >= 0, read by antidiagonals, where T(n,k) = Sum_{j=0..n} k^j * binomial(n,j)^3.

Original entry on oeis.org

1, 1, 1, 1, 2, 1, 1, 3, 10, 1, 1, 4, 21, 56, 1, 1, 5, 34, 171, 346, 1, 1, 6, 49, 352, 1521, 2252, 1, 1, 7, 66, 605, 3946, 14283, 15184, 1, 1, 8, 85, 936, 8065, 46744, 138909, 104960, 1, 1, 9, 106, 1351, 14346, 113525, 573616, 1385163, 739162, 1, 1, 10, 129, 1856, 23281, 231876, 1656145, 7217536, 14072193, 5280932, 1
Offset: 0

Views

Author

Seiichi Manyama, Jul 10 2020

Keywords

Comments

Column k is the diagonal of the rational function 1 / (1 + y + z + x*y + y*z + k*z*x + (k+1)*x*y*z).
Column k is the diagonal of the rational function 1 / ((1-x)*(1-y)*(1-z) - k*x*y*z).

Examples

			Square array begins:
  1,    1,     1,     1,      1,      1, ...
  1,    2,     3,     4,      5,      6, ...
  1,   10,    21,    34,     49,     66, ...
  1,   56,   171,   352,    605,    936, ...
  1,  346,  1521,  3946,   8065,  14346, ...
  1, 2252, 14283, 46744, 113525, 231876, ...

Crossrefs

Columns k=0-6 give: A000012, A000172, A206178, A206180, A216483, A216636, A216698.
Main diagonal gives A241247.
Cf. A307883, A336179, A336187.

Programs

  • Mathematica
    Unprotect[Power]; 0^0 = 1; T[n_, k_] := Sum[k^j * Binomial[n, j]^3, {j, 0, n}]; Table[T[k, n - k], {n, 0, 10}, {k, 0, n}] // Flatten (* Amiram Eldar, Jul 11 2020 *)
Showing 1-5 of 5 results.

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

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