A206180 -id:A206180 - cp's OEIS frontend

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

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

Offset: 0

Paul D. Hanna, Feb 04 2012

nonn

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

			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 +...

Vincenzo Librandi, Table of n, a(n) for n = 0..200
Vaclav Kotesovec, Asymptotic of a sums of powers of binomial coefficients * x^k, 2012.

Cf. A206177, A000172, A206180, A216483, A216636.

Related to diagonal of rational functions: A268545-A268555.

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)}
```

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

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

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

Vaclav Kotesovec, Sep 11 2012

nonn

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

Vincenzo Librandi, Table of n, a(n) for n = 0..200
V. Kotesovec, Asymptotic of a sums of powers of binomial coefficients * x^k, 2012.

Cf. A206178, A206180, A216636.

Table[Sum[Binomial[n,k]^3*4^k,{k,0,n}],{n,0,20}]

a(n) = sum(k=0, n, binomial(n,k)^3 * 4^k); \\ Gheorghe Coserea, Jul 01 2018

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

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

Minor edits by Vaclav Kotesovec, Mar 31 2014

A216636 a(n) = Sum_{k=0..n} binomial(n,k)^3 * 5^k.

Original entry on oeis.org

1, 6, 66, 936, 14346, 231876, 3885456, 66767616, 1169068986, 20769386796, 373277526876, 6772297456656, 123834925330416, 2279408745325536, 42194656181618496, 784905308800229376, 14663340953943086106, 274968958499402854716, 5173516852494573136836

Offset: 0

Vaclav Kotesovec, Sep 11 2012

nonn

Diagonal of rational function 1/(1 + y + z + x*y + y*z + 5*x*z + 6*x*y*z). - Gheorghe Coserea, Jul 01 2018

Diagonal of rational function 1 / ((1-x)*(1-y)*(1-z) - 5*x*y*z). - Seiichi Manyama, Jul 11 2020

Vincenzo Librandi, Table of n, a(n) for n = 0..200
Vaclav Kotesovec, Asymptotic of a sums of powers of binomial coefficients * x^k, 2012.

Cf. A206178, A206180, A216483.

Table[Sum[Binomial[n,k]^3*5^k,{k,0,n}],{n,0,20}]

a(n) = sum(k=0, n, binomial(n,k)^3 * 5^k); \\ Gheorghe Coserea, Jul 01 2018

A216636 = lambda n: hypergeometric([-n,-n,-n],[1,1], -5)
[Integer(A216636(n).n(100)) for n in (0..18)] # Peter Luschny, Sep 23 2014

Recurrence: (n+3)^2*(3*n+4)*a(n+3) = 6*(9*n^3+57*n^2+116*n+74)*a(n+2) + 3*(27*n^3+144*n^2+261*n+160)*a(n+1) + 216*(3*n+7)*(n+1)^2*a(n).
a(n) ~ (1 + 5^(1/3))^(3*n+2) / (2*sqrt(3)*5^(1/3)*Pi*n). - Vaclav Kotesovec, Sep 19 2012, simplified Apr 24 2025
G.f.: hypergeom([1/3, 2/3],[1],5*27*x^2/(1-6*x)^3)/(1-6*x). - Mark van Hoeij, May 02 2013
a(n) = hypergeom([-n,-n,-n],[1,1], -5). - Peter Luschny, Sep 23 2014
G.f. y=A(x) satisfies: x*(3*x + 1)*(216*x^3 + 27*x^2 + 18*x - 1)*y'' + (1944*x^4 + 1026*x^3 + 135*x^2 + 36*x - 1)*y' + 6*(108*x^3 + 69*x^2 + 2*x + 1)*y. - Gheorghe Coserea, Jul 01 2018

Minor edits by Vaclav Kotesovec, Mar 31 2014

A336182 a(n) = Sum_{k=0..n} (-3)^k * binomial(n,k)^3.

Original entry on oeis.org

1, -2, -14, 136, 106, -8492, 35344, 395008, -4547462, -4838372, 365951356, -1601617712, -19715085584, 233866581856, 285409397056, -20406741254144, 90043530872218, 1169513126877676, -13961261999882204, -18779832792734384, 1270510266589738636, -5584024444211882792

Offset: 0

Seiichi Manyama, Jul 10 2020

sign

Diagonal of the rational function 1 / (1 + y + z + x*y + y*z - 3*z*x - 2*x*y*z).

Diagonal of the rational function 1 / ((1-x)*(1-y)*(1-z) + 3*x*y*z).

Robert Israel, Table of n, a(n) for n = 0..1020

Column k=3 of A336179.

Cf. A206180.

f:= gfun:-rectoproc({(24*n^3 + 176*n^2 + 416*n + 320)*a(n + 1) + (279*n^3 + 2325*n^2 + 6382*n + 5776)*a(n + 2) + (18*n^3 + 168*n^2 + 514*n + 512)*a(n + 3) + (3*n^3 + 31*n^2 + 104*n + 112)*a(n + 4), a(0) = 1, a(1) = -2, a(2) = -14, a(3) = 136},a(n),remember):
map(f, [$0..30]); # Robert Israel, Jul 12 2020

a[n_] := Sum[(-3)^k * Binomial[n, k]^3, {k, 0, n}]; Array[a, 22, 0] (* Amiram Eldar, Jul 11 2020 *)

{a(n) = sum(k=0, n, (-3)^k*binomial(n,k)^3)}

From Robert Israel, Jul 12 2020: (Start)
a(n) = hypergeom([-n,-n,-n],[1,1],3).
(24*n^3 + 176*n^2 + 416*n + 320)*a(n + 1) + (279*n^3 + 2325*n^2 + 6382*n + 5776)*a(n + 2) + (18*n^3 + 168*n^2 + 514*n + 512)*a(n + 3) + (3*n^3 + 31*n^2 + 104*n + 112)*a(n + 4)=0. (End)

A336187 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)^k.

Original entry on oeis.org

1, 1, 1, 1, 2, 1, 1, 3, 4, 1, 1, 4, 13, 8, 1, 1, 5, 34, 63, 16, 1, 1, 6, 81, 352, 321, 32, 1, 1, 7, 186, 1685, 3946, 1683, 64, 1, 1, 8, 421, 7416, 38401, 46744, 8989, 128, 1, 1, 9, 946, 30835, 328146, 963525, 573616, 48639, 256, 1, 1, 10, 2113, 122816, 2590225, 16971876, 25346385, 7217536, 265729, 512, 1

Offset: 0

Seiichi Manyama, Jul 11 2020

Column k is the diagonal of the rational function 1 / (Product_{j=1..k} (1-x_j) - k * Product_{j=1..k} x_j) for k>0.

			Square array begins:
  1,  1,    1,     1,      1,        1, ...
  1,  2,    3,     4,      5,        6, ...
  1,  4,   13,    34,     81,      186, ...
  1,  8,   63,   352,   1685,     7416, ...
  1, 16,  321,  3946,  38401,   328146, ...
  1, 32, 1683, 46744, 963525, 16971876, ...

Columns k=0-3 give: A000012, A000079, A001850, A206180.
Main diagonal gives A336188.
Cf. A307883, A336163.

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

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

Vaclav Kotesovec, Sep 15 2012

nonn

Vincenzo Librandi, Table of n, a(n) for n = 0..200
V. Kotesovec, Asymptotic of a sums of powers of binomial coefficients * x^k, 2012

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

Table[Sum[Binomial[n, k]^3*6^k, {k, 0, n}], {n, 0, 25}]

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

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

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

Seiichi Manyama, Jul 10 2020

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).

			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, ...

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

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 *)

A206179 G.f.: exp( Sum_{n>=1} x^n/n * Sum_{k=0..n} binomial(n,k)^3 * 3^k ).

Original entry on oeis.org

1, 4, 25, 196, 1747, 16996, 175936, 1907224, 21423385, 247515796, 2925668236, 35239199704, 431207470105, 5347823877172, 67093724913313, 850241358959044, 10869754843088962, 140045452765874704, 1816842996684686656, 23716478425653945472, 311314468637807994391

Offset: 0

Paul D. Hanna, Feb 04 2012

nonn

Logarithmic derivative yields A206180.

			G.f.: A(x) = 1 + 4*x + 25*x^2 + 196*x^3 + 1747*x^4 + 16996*x^5 +...
where
log(A(x)) = 4*x + 34*x^2/2 + 352*x^3/3 + 3946*x^4/4 + 46744*x^5/5 + 573616*x^6/6 +...+ A206180(n)*x^n/n +...

Cf. A206180, A206177.

{a(n)=polcoeff(exp(sum(m=1,n,x^m/m*sum(k=0,m,binomial(m,k)^3*3^k))+x*O(x^n)),n)}
for(n=0,41,print1(a(n),", "))

cp's OEIS Frontend

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

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

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A216636 a(n) = Sum_{k=0..n} binomial(n,k)^3 * 5^k.

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A336182 a(n) = Sum_{k=0..n} (-3)^k * binomial(n,k)^3.

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A336187 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)^k.

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

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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.

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