A037966 -id:A037966 - cp's OEIS frontend

A037972 a(n) = n^2(n+1)binomial(2*n-2, n-1)/2.

0, 1, 12, 108, 800, 5250, 31752, 181104, 988416, 5212350, 26741000, 134132856, 660284352, 3199016548, 15288882000, 72209880000, 337535723520, 1563410094390, 7182839945160, 32761238433000, 148450107960000, 668693511305820, 2995943329133040, 13356820221694560

Offset: 0

N. J. A. Sloane

nonn

Identity (3.78), S_{3}, in H. W. Gould, Combinatorial Identities, Morgantown, 1972, page 31.

Seiichi Manyama, Table of n, a(n) for n = 0..1000

Cf. A000108, A000217, A037965, A037966, A074334, A329913, A329444.

[n^2*(n+1)*Binomial(2*n-2, n-1)/2: n in [0..30]]; // G. C. Greubel, Jun 22 2022

Table[n^2*Binomial[n+1,2]*CatalanNumber[n-1], {n,0,30}] (* G. C. Greubel, Jun 22 2022 *)

{a(n) = n^2*(n+1)*binomial(2*n-2, n-1)/2} \\ Seiichi Manyama, Aug 09 2020

[n^2*binomial(n+1,2)*catalan_number(n-1) for n in (0..30)] # G. C. Greubel, Jun 22 2022

a(n) = Sum_{k=0..n} k^3*(C(n,k))^2. [heruneedollar (heruneedollar(AT)gmail.com), Mar 20 2010]
a(n) = A000217(n)*A037965(n). - R. J. Mathar, Jul 26 2015
(n-1)^2*a(n) = 2*(11*n-16)*a(n-1) + 8*n*(2*n-5)*a(n-2). - R. J. Mathar, Oct 20 2015
(n-1)^3*a(n) = 2*n*(n+1)*(2*n-3)*a(n-1). - R. J. Mathar, Oct 20 2015
G.f.: x * (1 - 2*x + 10*x^2 - 12*x^3) / (1 - 4*x)^(7/2). - Ilya Gutkovskiy, Nov 17 2021

A329444 The sixth moments of the squared binomial coefficients; a(n) = Sum_{m=0..n} m^6*binomial(n, m)^2.

Original entry on oeis.org

0, 1, 68, 1314, 18080, 197350, 1836792, 15233316, 115776768, 821760390, 5520171800, 35438827996, 219038609088, 1310833221724, 7629754810160, 43348888067400, 241117582878720, 1316197491501510, 7065439665315480, 37362065079691500, 194909773207512000, 1004374157379474420

Offset: 0

Nikita D. Gogin, Nov 16 2019

nonn

H. W. Gould, Combinatorial Identities, 1972. (See formulas 3.77, 3.78, and 3.79 on page 31.)

Seiichi Manyama, Table of n, a(n) for n = 0..1000

Cf. A037966, A037972, A074334, A294486, A329521, A329913.

[(&+[Binomial(n,k)^2*k^6: k in [0..n]]): n in [0..30]]; // G. C. Greubel, Jun 23 2022

Table[Sum[m^6*(Binomial[n, m])^2, {m, 0, n}], {n, 21}]

a(n) = sum(m=0, n, m^6*binomial(n, m)^2); \\ Jinyuan Wang, Nov 23 2019

[n^3*(n+1)*(n^6+3*n^5-13*n^4-15*n^3+30*n^2+8*n-2)*catalan_number(n)/(8*(2*n-1)*(2*n-3)*(2*n-5)) for n in (0..30)] # G. C. Greubel, Jun 23 2022

a(n) = binomial(2*n, n) * n^3*(n^6 + 3*n^5 - 13*n^4 - 15*n^3 + 30*n^2 + 8*n - 2)/(8*(2*n-1)*(2*n-3)*(2*n-5)).

G.f.: x*(1 + 42*x - 168*x^2 + 1648*x^3 - 7608*x^4 + 18144*x^5 - 19376*x^6 - 1440*x^7 + 14400*x^8)/((1-4*x)^6*sqrt(1-4*x)). - G. C. Greubel, Jun 23 2022

A074334 a(n) = Sum_{r=1..n} r^4*binomial(n,r)^2.

Original entry on oeis.org

0, 1, 20, 234, 2144, 16750, 117432, 761460, 4654848, 27173718, 152867000, 834212236, 4438175040, 23108423884, 118111709744, 594059985000, 2946077521920, 14429322555750, 69892354873080, 335194270938780, 1593211647720000, 7511501237722020, 35153884344493200

Offset: 0

Paul Boddington, Mar 05 2003

H. W. Gould, Combinatorial Identities, 1972. (See formulas 3.77, 3.78, and 3.79 on page 31.)

Andrew Howroyd, Table of n, a(n) for n = 0..200
Nikita Gogin and Mika Hirvensalo, On the Moments of Squared Binomial Coefficients, (2020).

Cf. A000108 (Catalan numbers).

Cf. A000984, A002457, A037966, A037972, A329444, A329913.

[n le 1 select n else n^2*(n^3+n^2-3*n-1)*Catalan(n-2): n in [0..30]]; // G. C. Greubel, Jun 23 2022

Total/@Table[r^4 Binomial[n,r]^2,{n,0,20},{r,n}] (* Harvey P. Dale, Dec 04 2017 *)
Table[n^2*(n^3+n^2-3*n-1)*CatalanNumber[n-2] -Boole[n==1], {n,0,30}] (* G. C. Greubel, Jun 23 2022 *)

vector(30, n, n--; sum(k=1, n, k^4*binomial(n,k)^2)) \\ Michel Marcus, Aug 19 2015

[n^2*(n^3+n^2-3*n-1)*catalan_number(n-2) for n in (0..30)] # G. C. Greubel, Jun 23 2022

For n>1 a(n) = n^2*(n^3+n^2-3*n-1)*C(n-2). Here C(n-2) = binomial(2*n-4, n-2)/(n-1) is a Catalan number.
From G. C. Greubel, Jun 23 2022: (Start)
a(n) = (n^2*(n^3 + n^2 - 3*n -1)/(2*(2*n-3)))*binomial(2*n-2, n-1).
G.f.: x*(1 + 2*x + 32*x^3 - 128*x^4 + 144*x^5)/(1-4*x)^(9/2).
E.g.f.: x*exp(2*x)*( (1+2*x)*(1 +6*x +4*x^2)*BesselI(0, 2*x) + 2*x*(2 + 7*x + 4*x^2)*BesselI(1, 2*x) ). (End)
D-finite with recurrence (n-1)*(39*n-106)*a(n) +4*(-38*n^2+n+290)*a(n-1) +4*(100*n^2-784*n+1145)*a(n-2) -64*(13*n+4)*(2*n-9)*a(n-3)=0. - R. J. Mathar, Sep 13 2024

Terms a(18) and beyond from Andrew Howroyd, Jan 16 2020

A141611 Triangle read by rows: T(n, k) = (n-k+1)(k+1)binomial(n, k).

Original entry on oeis.org

1, 2, 2, 3, 8, 3, 4, 18, 18, 4, 5, 32, 54, 32, 5, 6, 50, 120, 120, 50, 6, 7, 72, 225, 320, 225, 72, 7, 8, 98, 378, 700, 700, 378, 98, 8, 9, 128, 588, 1344, 1750, 1344, 588, 128, 9, 10, 162, 864, 2352, 3780, 3780, 2352, 864, 162, 10, 11, 200, 1215, 3840, 7350, 9072, 7350, 3840, 1215, 200, 11

Offset: 0

Roger L. Bagula and Gary W. Adamson, Aug 22 2008

Read as a square array, this array factorizes as M*transpose(M), where M = ( k*binomial(n, k) )A003506(n,k).%20-%20_Peter%20Bala">{n,k>=1} = A003506(n,k). - _Peter Bala, Mar 06 2017

			Triangle begins as:
   1;
   2,   2;
   3,   8,    3;
   4,  18,   18,    4;
   5,  32,   54,   32,    5;
   6,  50,  120,  120,   50,    6;
   7,  72,  225,  320,  225,   72,    7;
   8,  98,  378,  700,  700,  378,   98,    8;
   9, 128,  588, 1344, 1750, 1344,  588,  128,    9;
  10, 162,  864, 2352, 3780, 3780, 2352,  864,  162,  10;
  11, 200, 1215, 3840, 7350, 9072, 7350, 3840, 1215, 200, 11;
  ...
From _Peter Bala_, Mar 06 2017: (Start)
Factorization as a square array
  /1         \ /1  2  3  4...\ /1  2   3   4...\
  |2  2      | |   2  6 12...| |2  8  12  32...|
  |3  6  3   |*|      3 12...|=|3 18  54 120...|
  |4 12 12 4 | |         4...| |4 32 120 320...|
  |...       | |             | |...            |
(End)

Indranil Ghosh, Table of n, a(n) for n = 1..5151 (Rows 0..100 of triangle, flattened)

Cf. A003506, A007466 (row sums), A037966, A085373.

A141611:= func< n,k | (k+1)*(n-k+1)*Binomial(n,k) >;
[A141611(n,k): k in [0..n], n in [0..14]]; // G. C. Greubel, Sep 22 2024

T[n_, m_]:= (n-m+1)*(m+1)*Binomial[n,m];
Table[T[n, m], {n,0,12}, {m,0,n}]//Flatten

T(n,m)=(n - m + 1)*(m + 1)*binomial(n, m) \\ Charles R Greathouse IV, Feb 15 2017

def A141611(n,k): return (k+1)*(n-k+1)*binomial(n,k)
flatten([[A141611(n,k) for k in range(n+1)] for n in range(15)]) # G. C. Greubel, Sep 22 2024

T(n, k) = (k+1)*(n-k+1)*binomial(n,k).
Sum_{k=0..n} T(n, k) = A007466(n+1) (row sums).
O.g.f.: (1 - (1 + t)*x + 2*t*x^2)/(1 - (1 + t)*x)^3 = 1 + (2 + 2*t)*x + (3 + 8*t + 3*t^2)*x^2 + (4 + 18*t + 18*t^2 + 4*t^3)*x^3 + .... - Peter Bala, Mar 06 2017
From G. C. Greubel, Sep 22 2024: (Start)
T(2*n, n) = A037966(n+1).
T(2*n-1, n) = 2*A085373(n-1), for n >= 1.
Sum_{k=0..n} (-1)^k*T(n, k) = A000007(n) - 2*[n=2]. (End)

Offset corrected by G. C. Greubel, Sep 22 2024

A329913 The fifth moments of the squared binomial coefficients; a(n) = Sum_{m=0..n} m^5*binomial(n, m)^2.

Original entry on oeis.org

0, 1, 36, 540, 6080, 56250, 455112, 3342192, 22809600, 146988270, 904475000, 5358254616, 30750385536, 171773279860, 937514244240, 5014575000000, 26351064760320, 136319273714070, 695429503781400, 3503580441563400, 17452918098000000, 86055711108818220

Offset: 0

Nikita D. Gogin, Nov 24 2019

nonn

H. W. Gould, Combinatorial Identities, 1972. (See formulas 3.77, 3.78, and 3.79 on page 31.)

Robert Israel, Table of n, a(n) for n = 0..1638

Cf. A074334, A294486.

Cf. A000984, A002457, A037966, A037972, A074334, A329444.

[(&+[Binomial(n,k)^2*k^5: k in [0..n]]): n in [0..30]]; // G. C. Greubel, Jun 23 2022

seq( binomial(2*n,n)*n^4*(n^3 + 3*n^2 - 3*n - 5)/((16*n-8)*(2*n-3)),n=0..30); # Robert Israel, Jan 26 2020

Table[Sum[m^5*(Binomial[n, m])^2, {m, 0, n}], {n, 21}]

a(n) = sum(k=0, n, k^5*binomial(n, k)^2); \\ Michel Marcus, Nov 24 2019

[n^4*(n+1)*(n^3+3*n^2-3*n-5)/(8*(2*n-1)*(2*n-3))*catalan_number(n) for n in (0..30)] # G. C. Greubel, Jun 23 2022

a(n) = binomial(2*n,n)*n^4*(n^3 + 3*n^2 - 3*n - 5)/(8*(2*n-1)*(2*n-3)).
G.f.: x*(1 + 14*x - 54*x^2 + 404*x^3 - 1544*x^4 + 2880*x^5 - 2160*x^6)/(1-4*x)^(11/2). - Stefano Spezia, Jan 03 2020
(-12960 + 8640*n)*a(n) + (7200 - 13680*n)*a(n + 1) + (3920 + 9056*n)*a(n + 2) + (-4184 - 3160*n)*a(n + 3) + (1404 + 620*n)*a(n + 4) + (-584 - 110*n)*a(n + 5) + (14 + 10*n)*a(n + 6) + (n + 6)*a(n + 7) = 0. - Robert Israel, Jan 26 2020

A336828 a(n) = Sum_{k=0..n} binomial(n,k)^2 * k^n.

Original entry on oeis.org

1, 1, 8, 108, 2144, 56250, 1836792, 71799504, 3269445888, 169974711630, 9934458411800, 644825382429096, 46022332032100800, 3582265183110626740, 302002255041807372080, 27413749834141448520000, 2665789990569658618398720, 276477318687585566522176470

Offset: 0

Ilya Gutkovskiy, Aug 05 2020

nonn

Seiichi Manyama, Table of n, a(n) for n = 0..343

Cf. A000984, A002457, A037966, A037972, A072034, A074334, A187021, A329444, A329913, A336214, A341815.

Join[{1}, Table[Sum[Binomial[n, k]^2 k^n, {k, 0, n}], {n, 1, 17}]]

a(n) = sum(k=0, n, binomial(n, k)^2*k^n); \\ Michel Marcus, Aug 05 2020

a(n) ~ c * d^n * (n-1)!, where d = (1 + 2*LambertW(exp(-1/2)/2)) / (4*LambertW(exp(-1/2)/2)^2) = 6.476217542109791521947605963458797355564... and c = 0.21617818094152997942246965143216887599763501682724844713834495... - Vaclav Kotesovec, Feb 20 2021

A174126 Triangle T(n, k) = (n-k)^2 * binomial(n-1, k-1)^2 with T(n, 0) = T(n, n) = 1, read by rows.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 4, 4, 1, 1, 9, 36, 9, 1, 1, 16, 144, 144, 16, 1, 1, 25, 400, 900, 400, 25, 1, 1, 36, 900, 3600, 3600, 900, 36, 1, 1, 49, 1764, 11025, 19600, 11025, 1764, 49, 1, 1, 64, 3136, 28224, 78400, 78400, 28224, 3136, 64, 1, 1, 81, 5184, 63504, 254016, 396900, 254016, 63504, 5184, 81, 1

Offset: 0

Roger L. Bagula, Mar 09 2010

This triangle sequence is part of a class of triangles defined by T(n, k, q) = (n-k)^q * binomial(n-1, k-1)^q with T(n, 0) = T(n, n) = 1 and have row sums Sum_{k=0..n} T(n, k, q) = 2 - [n=0] + Sum_{k=1..n-1} k^q * binomial(n-1, k)^q. - G. C. Greubel, Feb 10 2021

			Triangle begins as:
  1;
  1,  1;
  1,  1,    1;
  1,  4,    4,     1;
  1,  9,   36,     9,      1;
  1, 16,  144,   144,     16,      1;
  1, 25,  400,   900,    400,     25,      1;
  1, 36,  900,  3600,   3600,    900,     36,     1;
  1, 49, 1764, 11025,  19600,  11025,   1764,    49,    1;
  1, 64, 3136, 28224,  78400,  78400,  28224,  3136,   64,  1;
  1, 81, 5184, 63504, 254016, 396900, 254016, 63504, 5184, 81, 1;

G. C. Greubel, Rows n = 0..100 of the triangle, flattened

Cf. A000108, A037966.

Cf. A155865 (q=1), this sequence (q=2), A174127 (q=3).

T:= func< n,k,q | k eq 0 or k eq n select 1 else (n-k)^q*Binomial(n-1,k-1)^q >;
[T(n,k,2): k in [0..n], n in [0..12]]; // G. C. Greubel, Feb 10 2021

(* First program *)
c[n_]:= If[n<2, 1, Product[(i-1)^2, {i,2,n}]];
T[n_, k_]:= c[n]/(c[k]*c[n-k]);
Table[T[n, k], {n,0,12}, {k,0,n}]//Flatten
(* Second program *)
T[n_, k_, q_]:= If[k==0 || k==n, 1, (n-k)^q*Binomial[n-1, k-1]^q];
Table[T[n,k,2], {n,0,12}, {k,0,n}]//Flatten (* G. C. Greubel, Feb 10 2021 *)

def T(n,k,q): return 1 if (k==0 or k==n) else (n-k)^q*binomial(n-1,k-1)^q
flatten([[T(n,k,2) for k in (0..n)] for n in (0..12)]) # G. C. Greubel, Feb 10 2021

Let c(n) = Product_{i=2..n} (i-1)^2 for n > 2 otherwise 1. The number triangle is given by T(n, k) = c(n)/(c(k)*c(n-k)).
From G. C. Greubel, Feb 10 2021: (Start)
T(n, k) = (n-k)^2 * binomial(n-1, k-1)^2 with T(n, 0) = T(n, n) = 1.
Sum_{k=0..n} T(n, k) = 2 + A037966(n-1) - [n=0] = 2 + (n-1)^3*C_{n-2} - [n=0], where C_{n} are the Catalan numbers (A000108) and [] is the Iverson bracket. (End)

Edited by G. C. Greubel, Feb 10 2021

A349427 a(n) = ((n+1)^2 - 2) * binomial(2*n-2,n-1) / 2.

Original entry on oeis.org

0, 1, 7, 42, 230, 1190, 5922, 28644, 135564, 630630, 2892890, 13117676, 58903572, 262303132, 1159666900, 5094808200, 22259364120, 96773942790, 418882316490, 1805951924700, 7758285404100, 33221013445620, 141830949914940, 603876402587640, 2564713671647400

Offset: 0

Ilya Gutkovskiy, Nov 17 2021

Cf. A000108, A000217, A000984, A001700, A001793, A002457, A002544, A008865, A037966, A088218, A127736.

Table[((n + 1)^2 - 2) Binomial[2 n - 2, n - 1]/2, {n, 0, 24}]
nmax = 24; CoefficientList[Series[x (1 - x) (1 - 2 x)/(1 - 4 x)^(5/2), {x, 0, nmax}], x]

a(n) = ((n+1)^2 - 2) * binomial(2*n-2,n-1) / 2 \\ Andrew Howroyd, Nov 20 2021

G.f.: x * (1 - x) * (1 - 2*x) / (1 - 4*x)^(5/2).
E.g.f.: x * exp(2*x) * (2 * (1 + x) * BesselI(0,2*x) + (1 + 2*x) * BesselI(1,2*x)) / 2.
a(n) = n * ((n+1)^2 - 2) * Catalan(n-1) / 2.
a(n) = Sum_{k=0..n} binomial(n,k)^2 * A000217(k).
a(n) ~ 2^(2*n-3) * n^(3/2) / sqrt(Pi).
D-finite with recurrence (n-1)*(n^2-2)*a(n) -2*(2*n-3)*(n^2+2*n-1)*a(n-1)=0. - R. J. Mathar, Mar 06 2022

A336955 a(n) = Sum_{k=0..n} k^k * binomial(n, k)^2.

Original entry on oeis.org

1, 2, 9, 73, 849, 12651, 228493, 4836301, 117204545, 3196763983, 96842596701, 3224356269597, 116981406934417, 4591908332288837, 193846634326107701, 8755364023207809301, 421214258699748184321, 21500563181275847468503, 1160430732790051008442141, 66020998289431649938896445

Offset: 0

Seiichi Manyama, Aug 09 2020

nonn

Cf. A037966, A037972, A086331, A329913, A329444, A336828.

a[n_] := 1 + Sum[k^k * Binomial[n, k]^2, {k, 1, n}]; Array[a, 20, 0] (* Amiram Eldar, Aug 09 2020 *)

{a(n) = sum(k=0, n, k^k*binomial(n, k)^2)}

cp's OEIS Frontend

A037972 a(n) = n^2*(n+1)*binomial(2*n-2, n-1)/2.

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A329444 The sixth moments of the squared binomial coefficients; a(n) = Sum_{m=0..n} m^6*binomial(n, m)^2.

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A074334 a(n) = Sum_{r=1..n} r^4*binomial(n,r)^2.

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A141611 Triangle read by rows: T(n, k) = (n-k+1)*(k+1)*binomial(n, k).

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A329913 The fifth moments of the squared binomial coefficients; a(n) = Sum_{m=0..n} m^5*binomial(n, m)^2.

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A336828 a(n) = Sum_{k=0..n} binomial(n,k)^2 * k^n.

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A037972 a(n) = n^2(n+1)binomial(2*n-2, n-1)/2.

A141611 Triangle read by rows: T(n, k) = (n-k+1)(k+1)binomial(n, k).