A001788 -id:A001788 - cp's OEIS frontend

A276289 Expansion of x(1 + x)/(1 - 2x)^3.

0, 1, 7, 30, 104, 320, 912, 2464, 6400, 16128, 39680, 95744, 227328, 532480, 1232896, 2826240, 6422528, 14483456, 32440320, 72220672, 159907840, 352321536, 772800512, 1688207360, 3674210304, 7969177600, 17230200832, 37144756224, 79859548160, 171261820928, 366414397440

Offset: 0

Ilya Gutkovskiy, Aug 27 2016

Binomial transform of pentagonal numbers (A000326).
More generally, the binomial transform of k-gonal numbers is n*Hypergeometric2F1(k/(k-2),1-n;2/(k-2);-1), where Hypergeometric2F1(a,b;c;x) is the hypergeometric function.
Coefficients in the hypergeometric series identity 1 - 7*x/(x + 6) + 30*x*(x - 1)/((x + 6)*(x + 8)) - 104*x*(x - 1)*(x - 2)/((x + 6)*(x + 8)*(x + 10)) + ... = 0, valid in the half-plane Re(x) > 0. Cf. A077616 and A084901. - Peter Bala, May 30 2019

G. C. Greubel, Table of n, a(n) for n = 0..1000
Eric Weisstein's World of Mathematics, Pentagonal Number
Index entries for linear recurrences with constant coefficients, signature (6,-12,8).

Cf. A000326, A084901.

Cf. A001793 (binomial transform of triangular numbers), A001788 (binomial transform of squares), A084899 (binomial transform of heptagonal numbers).

List([0..40], n-> 2^(n-3)*n*(3*n+1)); # G. C. Greubel, Jun 02 2019

[2^(n-3)*n*(3*n+1): n in [0..40]]; // G. C. Greubel, Jun 02 2019

a:=series(x*(1+x)/(1-2*x)^3,x=0,31): seq(coeff(a,x,n),n=0..40); # Paolo P. Lava, Mar 27 2019

LinearRecurrence[{6, -12, 8}, {0, 1, 7}, 40]
Table[2^(n - 3) n (3 n + 1), {n, 0, 40}]

concat(0, Vec(x*(1+x)/(1-2*x)^3 + O(x^40))) \\ Altug Alkan, Aug 27 2016

[2^(n-3)*n*(3*n+1) for n in (0..40)] # G. C. Greubel, Jun 02 2019

O.g.f.: x*(1 + x)/(1 - 2*x)^3.
E.g.f.: x*(2 + 3*x)*exp(2*x)/2.
a(n) = 6*a(n-1) - 12*a(n-2) + 8*a(n-3).
a(n) = Sum_{k = 0..n} binomial(n,k)*k*(3*k - 1)/2.
a(n) = 2^(n-3)*n*(3*n + 1).
Sum_{n>=1} 1/a(n) = 8*(-3*2^(1/3)*Hypergeometric2F1(1/3,1/3;4/3;-1) + 3 + log(2)) = 1.1906948190529335181687...

A374497 Expansion of 1/(1 - 4x - 4x^2)^(3/2).

Original entry on oeis.org

1, 6, 36, 200, 1080, 5712, 29792, 153792, 787680, 4009280, 20304768, 102405888, 514678528, 2579028480, 12890311680, 64283809792, 319954540032, 1589720712192, 7886437652480, 39069462835200, 193307835764736, 955361266917376, 4716674314223616, 23264437702656000

Offset: 0

Seiichi Manyama, Jul 09 2024

nonn

Cf. A001788, A002457, A025163.
Cf. A006139, A374511, A374513.
Cf. A071356.

a[n_]:= Sum[(2*k+1)*Binomial[2*k,k]*Binomial[k,n-k],{k,0,n}]; Array[a,24,0] (* Stefano Spezia, May 08 2025 *)

a(n) = binomial(n+2, 2)*sum(k=0, n\2, 2^(n-k)*binomial(n, 2*k)*binomial(2*k, k)/(k+1));

a(0) = 1, a(1) = 6; a(n) = (2*(2*n+1)*a(n-1) + 4*(n+1)*a(n-2))/n.
a(n) = binomial(n+2,2) * A071356(n).
a(n) = Sum_{k=0..n} (2*k+1) * binomial(2*k,k) * binomial(k,n-k). - Seiichi Manyama, Oct 19 2024
a(n) = ((n+2)/2) * Sum_{k=0..floor(n/2)} 2^(n-k) * binomial(n+1,n-2*k) * binomial(2*k+1,k). - Seiichi Manyama, Aug 20 2025

A058395 Square array read by antidiagonals. Based on triangular numbers (A000217) with each term being the sum of 2 consecutive terms in the previous row.

Original entry on oeis.org

1, 0, 1, 3, 1, 1, 0, 3, 2, 1, 6, 3, 4, 3, 1, 0, 6, 6, 6, 4, 1, 10, 6, 9, 10, 9, 5, 1, 0, 10, 12, 15, 16, 13, 6, 1, 15, 10, 16, 21, 25, 25, 18, 7, 1, 0, 15, 20, 28, 36, 41, 38, 24, 8, 1, 21, 15, 25, 36, 49, 61, 66, 56, 31, 9, 1, 0, 21, 30, 45, 64, 85, 102, 104, 80, 39, 10, 1, 28, 21, 36, 55, 81, 113, 146, 168, 160, 111, 48, 11, 1

Offset: 0

Henry Bottomley, Nov 24 2000

Changing the formula by replacing T(2n, 0) = T(n, 3) with T(2n, 0) = T(n, m) for some other value of m would change the generating function to the coefficient of x^n in expansion of (1 + x)^k / (1 - x^2)^m. This would produce A058393, A058394, A057884 (and effectively A007318).

			The array T(n, k) starts:
[0] 1, 0,  3,   0,   6,   0,  10,    0,   15,    0, ...
[1] 1, 1,  3,   3,   6,   6,  10,   10,   15,   15, ...
[2] 1, 2,  4,   6,   9,  12,  16,   20,   25,   30, ...
[3] 1, 3,  6,  10,  15,  21,  28,   36,   45,   55, ...
[4] 1, 4,  9,  16,  25,  36,  49,   64,   81,  100, ...
[5] 1, 5, 13,  25,  41,  61,  85,  113,  145,  181, ...
[6] 1, 6, 18,  38,  66, 102, 146,  198,  258,  326, ...
[7] 1, 7, 24,  56, 104, 168, 248,  344,  456,  584, ...
[8] 1, 8, 31,  80, 160, 272, 416,  592,  800, 1040, ...
[9] 1, 9, 39, 111, 240, 432, 688, 1008, 1392, 1840, ...

Rows are A000217 with zeros, A008805, A002620, A000217, A000290, A001844, A005899.
Columns are A000012, A001477, A016028.
Diagonals include A058396, A049611, A001793, A001788, A055580, A055581, A055582.
The triangle A055252 also appears in half of the array.

gf := n -> (1 + x)^n / (1 - x^2)^3: ser := n -> series(gf(n), x, 20):
seq(lprint([n], seq(coeff(ser(n), x, k), k = 0..9)), n = 0..9); # Peter Luschny, Apr 12 2023

T[0, k_] := If[OddQ[k], 0, (k+2)(k+4)/8];
T[n_, k_] := T[n, k] = If[k == 0, 1, T[n-1, k-1] + T[n-1, k]];
Table[T[n-k, k], {n, 0, 12}, {k, n, 0, -1}] // Flatten (* Jean-François Alcover, Apr 13 2023 *)

T(n, k) = T(n-1, k-1) + T(n, k-1) with T(0, k) = 1, T(2*n, 0) = T(n, 3) and T(2*n + 1, 0) = 0. Coefficient of x^n in expansion of (1 + x)^k / (1 - x^2)^3.

A059299 Triangle of idempotent numbers (version 3), T(n, k) = binomial(n, k) * (n - k)^k.

Original entry on oeis.org

1, 1, 0, 1, 2, 0, 1, 6, 3, 0, 1, 12, 24, 4, 0, 1, 20, 90, 80, 5, 0, 1, 30, 240, 540, 240, 6, 0, 1, 42, 525, 2240, 2835, 672, 7, 0, 1, 56, 1008, 7000, 17920, 13608, 1792, 8, 0, 1, 72, 1764, 18144, 78750, 129024, 61236, 4608, 9, 0, 1, 90, 2880, 41160

Offset: 0

N. J. A. Sloane, Jan 25 2001

			Triangle begins:
1,
1,  0,
1,  2,   0,
1,  6,   3,    0,
1, 12,  24,    4,    0,
1, 20,  90,   80,    5,   0,
1, 30, 240,  540,  240,   6, 0,
1, 42, 525, 2240, 2835, 672, 7, 0,
...

L. Comtet, Advanced Combinatorics, Reidel, 1974, p. 91, #43 and p. 135, [3i'].

G. C. Greubel, Table of n, a(n) for the first 50 rows, flattened

There are 4 versions: A059297-A059300.
Diagonals give A001788, A036216, A040075, A050982, A002378, 3*A002417, etc.
Row sums are A000248.

/* As triangle: */ [[Binomial(n,k)*(n-k)^k: k in [0..n]]: n in [0.. 15]]; // Vincenzo Librandi, Aug 22 2015

T := (n, k) -> binomial(n, k) * (n - k)^k:
for n from 0 to 9 do seq(T(n, k), k = 0..n) od;

t[n_, k_] := Binomial[n, k]*(n - k)^k; Prepend[Flatten@Table[t[n, k], {n, 10}, {k, 0, n}], 1] (* Arkadiusz Wesolowski, Mar 23 2013 *)

concat([1], for(n=0, 25, for(k=0, n, print1(binomial(n,k)*(n-k)^k, ", ")))) \\ G. C. Greubel, Jan 05 2017

Name corrected by Peter Luschny, Nov 12 2023

A129532 3n(n-1)4^(n-2).

Original entry on oeis.org

0, 0, 6, 72, 576, 3840, 23040, 129024, 688128, 3538944, 17694720, 86507520, 415236096, 1962934272, 9160359936, 42278584320, 193273528320, 876173328384, 3942779977728, 17626545782784, 78340203479040, 346346162749440

Offset: 0

Emeric Deutsch, Apr 22 2007

nonn

Number of inversions in all 4-ary words of length n on {0,1,2,3}. Example: a(2)=6 because each of the words 10,20,30,21,31,32 has one inversion and the words 00,01,02,03,11,12,13,22,23,33 have no inversions. a(n)=Sum(k*A129531(n,k),k>=0). a(n)=6*A038845(n-2).

Index entries for linear recurrences with constant coefficients, signature (12,-48,64).

Cf. A038845, A129531, A001788, A129530.

Maple
```
seq(3*n*(n-1)*4^(n-2),n=0..25);
```

Table[3n(n-1)4^(n-2),{n,0,30}] (* or *) LinearRecurrence[{12,-48,64},{0,0,6},30] (* Harvey P. Dale, May 25 2018 *)

G.f.=6x^2/(1-4x)^3.

A130812 If X_1,...,X_n is a partition of a 2n-set X into 2-blocks then a(n) is equal to the number of 6-subsets of X containing none of X_i, (i=1,...n).

Original entry on oeis.org

64, 448, 1792, 5376, 13440, 29568, 59136, 109824, 192192, 320320, 512512, 792064, 1188096, 1736448, 2480640, 3472896, 4775232, 6460608, 8614144, 11334400, 14734720, 18944640, 24111360, 30401280, 38001600, 47121984, 57996288, 70884352, 86073856, 103882240

Offset: 6

Milan Janjic, Jul 16 2007

Number of n permutations (n>=6) of 3 objects u,v,z, with repetition allowed, containing n-6 u's. Example: if n=6 then n-6 =(0) zero u, a(1)=64. - Zerinvary Lajos, Aug 05 2008

a(n) is the number of 5-dimensional elements in an n-cross polytope where n>=6. - Patrick J. McNab, Jul 06 2015

Vincenzo Librandi, Table of n, a(n) for n = 6..1000
Milan Janjic, Two Enumerative Functions
Eric Weisstein's World of Mathematics, Cross Polytope

Cf. A038207, A000079, A001787, A001788, A001789, A003472, A054849, A002409, A054851, A140325, A140354, A046092, A130809, A130810, A130811. - Zerinvary Lajos, Aug 05 2008

[Binomial(2*n,6)+Binomial(n,2)*Binomial(2*n-4,2)- n*Binomial(2*n-2,4)-Binomial(n,3): n in [6..40]]; // Vincenzo Librandi, Jul 09 2015

a:=n->binomial(2*n,6)+binomial(n,2)*binomial(2*n-4,2)-n*binomial(2*n-2,4)-binomial(n,3);
seq(binomial(n,n-6)*2^6,n=6..32); # Zerinvary Lajos, Dec 07 2007
seq(binomial(n+5, 6)*2^6, n=1..22); # Zerinvary Lajos, Aug 05 2008

CoefficientList[Series[64/(1-x)^7,{x,0,30}],x] (* Vincenzo Librandi, Mar 21 2012 *)

a(n) = binomial(2*n,6) + binomial(n,2)*binomial(2*n-4,2) - n*binomial(2*n-2,4) - binomial(n,3).
a(n) = C(n,n-6)*2^6, n>=6. - Zerinvary Lajos, Dec 07 2007
G.f.: 64*x^6/(1-x)^7. - Colin Barker, Mar 20 2012

A158920 Binomial transform of A008805 (triangular numbers with repeats).

Original entry on oeis.org

1, 2, 6, 16, 41, 102, 248, 592, 1392, 3232, 7424, 16896, 38144, 85504, 190464, 421888, 929792, 2039808, 4456448, 9699328, 21037056, 45481984, 98041856, 210763776, 451936256, 966787072, 2063597568, 4395630592, 9344909312, 19830669312

Offset: 1

Gary W. Adamson, Mar 30 2009

			a(4) = 16 = (1, 3, 3, 1) dot (1, 1, 3, 3) = (1 + 3 + 9 + 3).

Harvey P. Dale, Table of n, a(n) for n = 1..1000
Milan Janjic and Boris Petkovic, A Counting Function, arXiv preprint arXiv:1301.4550 [math.CO], 2013. - From _N. J. A. Sloane_, Feb 13 2013
Milan Janjic and Boris Petkovic, A Counting Function Generalizing Binomial Coefficients and Some Other Classes of Integers, J. Int. Seq. 17 (2014) # 14.3.5.
Index entries for linear recurrences with constant coefficients, signature (6,-12,8).

Cf. A008805, A045891.

A000217 := proc(n) n*(n+1)/2 ; end: A008805 := proc(n) A000217( 1+floor(n/2) ) ; end: L := [seq(A008805(n), n=0..100)] ; read("transforms"); BINOMIAL(L) ; # R. J. Mathar, Apr 02 2009

Join[{1,2},LinearRecurrence[{6,-12,8},{6,16,41},30]] (* Harvey P. Dale, Feb 25 2012 *)

A007318 * (1, 1, 3, 3, 6, 6, 10, 10, 15, 15, ...) = binomial transform of triangular numbers A000217 with repeats.
From R. J. Mathar, Apr 02 2009: (Start)
G.f.: x*(x-1)^4/(1-2*x)^3.
a(n) = 6*a(n-1) - 12*a(n-2) + 8*a(n-3), n > 5. (End)
32*a(n) = 2^(n+1) + 3*A001787(n+1) + A001788(n+1), n>=3. - R. J. Mathar, Feb 25 2023

A167431 Riordan array (1-4x+4x^2, x(1-2x)).

Original entry on oeis.org

1, -4, 1, 4, -6, 1, 0, 12, -8, 1, 0, -8, 24, -10, 1, 0, 0, -32, 40, -12, 1, 0, 0, 16, -80, 60, -14, 1, 0, 0, 0, 80, -160, 84, -16, 1, 0, 0, 0, -32, 240, -280, 112, -18, 1, 0, 0, 0, 0, -192, 560, -448, 144, -20, 1, 0, 0, 0, 0, 64, -672, 1120, -672, 180, -22, 1

Offset: 0

Paul Barry, Nov 03 2009

Row sums are A167433. Diagonal sums are A167434. Central coefficients are (-1)^n*A001788(n+1).

Inverse is A167432.

			Triangle begins
1,
-4, 1,
4, -6, 1,
0, 12, -8, 1,
0, -8, 24, -10, 1,
0, 0, -32, 40, -12, 1,
0, 0, 16, -80, 60, -14, 1,
0, 0, 0, 80, -160, 84, -16, 1,
0, 0, 0, -32, 240, -280, 112, -18, 1,
0, 0, 0, 0, -192, 560, -448, 144, -20, 1,
0, 0, 0, 0, 64, -672, 1120, -672, 180, -22, 1

T(n,k)=T(n-1,k-1)-2*T(n-2,k-1), T(0,0)=1, T(1,0)=-4, T(2,0)=4, T(n,k)=0 if k<0 or if k>n. - Philippe Deléham, Nov 13 2013

A172242 Number of 10-D hypercubes in an n-dimensional hypercube.

Original entry on oeis.org

1, 22, 264, 2288, 16016, 96096, 512512, 2489344, 11202048, 47297536, 189190144, 722362368, 2648662016, 9372188672, 32133218304, 107110727680, 348109864960, 1105760747520, 3440144547840, 10501493882880, 31504481648640

Offset: 10

Zerinvary Lajos, Jan 29 2010

With a different offset, number of n-permutations (n>=8) of 3 objects: u, v, z with repetition allowed, containing exactly ten (10) u's.

Milan Janjic and Boris Petkovic, A Counting Function, arXiv 1301.4550 [math.CO], 2013.
Index entries for linear recurrences with constant coefficients, signature (22,-220,1320,-5280,14784,-29568,42240,-42240,28160,-11264,2048).

Cf. A001788, A001789, A003472, A038207, A054849, A002409, A140325, A054851, A140354.

Table[Binomial[n + 10, 10]*2^n, {n, 0, 22}]

[lucas_number2(n, 2, 0)*binomial(n,10)/2^10 for n in range(10, 31)] # Zerinvary Lajos, Feb 05 2010

a(n) = A038207(n,10).
a(n) = binomial(n,10)*2^(n-10). [Corrected by R. J. Mathar, Feb 21 2010]
G.f.: -x^10/(2*x-1)^11. - Colin Barker, Nov 11 2012
a(n) = Sum_{i=10..n} binomial(i,10)*binomial(n,i). Example: for n=15, a(15) = 1*3003 + 11*1365 + 66*455 + 286*105 + 1001*15 + 3003*1 = 96096. - Bruno Berselli, Mar 23 2018
From Amiram Eldar, Jan 07 2022: (Start)
Sum_{n>=10} 1/a(n) = 1879/126 - 20*log(2).
Sum_{n>=10} (-1)^n/a(n) = 393660*log(3/2) - 20111419/126. (End)

A291203 Number F(n,h,t) of forests of t labeled rooted trees with n vertices such that h is the maximum of 0 and the tree heights; triangle of triangles F(n,h,t), n>=0, h=0..n, t=0..n-h, read by layers, then by rows.

Original entry on oeis.org

1, 0, 1, 0, 0, 0, 1, 0, 2, 0, 0, 0, 0, 1, 0, 3, 6, 0, 6, 0, 0, 0, 0, 0, 1, 0, 4, 24, 12, 0, 36, 24, 0, 24, 0, 0, 0, 0, 0, 0, 1, 0, 5, 80, 90, 20, 0, 200, 300, 60, 0, 300, 120, 0, 120, 0, 0, 0, 0, 0, 0, 0, 1, 0, 6, 240, 540, 240, 30, 0, 1170, 3000, 1260, 120, 0, 3360, 2520, 360, 0, 2520, 720, 0, 720, 0

Offset: 0

Alois P. Heinz, Aug 20 2017

Positive elements in column t=1 give A034855.
Elements in rows h=0 give A023531.
Elements in rows h=1 give A059297.
Positive row sums per layer give A235595.
Positive column sums per layer give A061356.

			n h\t: 0   1   2  3  4 5 : A235595 : A061356          : A000272
-----+-------------------+---------+------------------+--------
0 0  : 1                 :         :                  : 1
-----+-------------------+---------+------------------+--------
1 0  : 0   1             :      1  :   .              :
1 1  : 0                 :         :   1              : 1
-----+-------------------+---------+------------------+--------
2 0  : 0   0   1         :      1  :   .   .          :
2 1  : 0   2             :      2  :   .              :
2 2  : 0                 :         :   2   1          : 3
-----+-------------------+---------+------------------+--------
3 0  : 0   0   0  1      :      1  :   .   .   .      :
3 1  : 0   3   6         :      9  :   .   .          :
3 2  : 0   6             :      6  :   .              :
3 3  : 0                 :         :   9   6   1      : 16
-----+-------------------+---------+------------------+--------
4 0  : 0   0   0  0  1   :      1  :   .   .   .  .   :
4 1  : 0   4  24 12      :     40  :   .   .   .      :
4 2  : 0  36  24         :     60  :   .   .          :
4 3  : 0  24             :     24  :   .              :
4 4  : 0                 :         :  64  48  12  1   : 125
-----+-------------------+---------+------------------+--------
5 0  : 0   0   0  0  0 1 :      1  :   .   .   .  . . :
5 1  : 0   5  80 90 20   :    195  :   .   .   .  .   :
5 2  : 0 200 300 60      :    560  :   .   .   .      :
5 3  : 0 300 120         :    420  :   .   .          :
5 4  : 0 120             :    120  :   .              :
5 5  : 0                 :         : 625 500 150 20 1 : 1296
-----+-------------------+---------+------------------+--------

Alois P. Heinz, Layers n = 0..48, flattened

Cf. A000007, A000012, A000142, A000272, A000551, A001477, A001788, A001854, A002378, A023531, A034855, A038154, A059297, A061356, A089946, A126804, A234953, A235595, A235596, A243014, A291204, A291336, A291529.

b:= proc(n, t, h) option remember; expand(`if`(n=0 or h=0, x^(t*n), add(
       binomial(n-1, j-1)*j*x^t*b(j-1, 0, h-1)*b(n-j, t, h), j=1..n)))
    end:
g:= (n, h)-> b(n, 1, h)-`if`(h=0, 0, b(n, 1, h-1)):
F:= (n, h, t)-> coeff(g(n, h), x, t):
seq(seq(seq(F(n, h, t), t=0..n-h), h=0..n), n=0..8);

b[n_, t_, h_] := b[n, t, h] = Expand[If[n == 0 || h == 0, x^(t*n), Sum[
     Binomial[n-1, j-1]*j*x^t*b[j-1, 0, h-1]*b[n-j, t, h], {j, 1, n}]]];
g[n_, h_] := b[n, 1, h] - If[h == 0, 0, b[n, 1, h - 1]];
F[n_, h_, t_] := Coefficient[g[n, h], x, t];
Table[Table[Table[F[n, h, t], {t, 0, n - h}], {h, 0, n}], {n, 0, 8}] // Flatten (* Jean-François Alcover, Mar 17 2022, after Alois P. Heinz *)

Sum_{i=0..n} F(n,i,n-i) = A243014(n) = 1 + A038154(n).
Sum_{d=0..n} Sum_{i=0..d} F(n,i,d-i) = A000272(n+1).
Sum_{h=0..n} Sum_{t=0..n-h} t * F(n,h,t) = A089946(n-1) for n>0.
Sum_{h=0..n} Sum_{t=0..n-h} (h+1) * F(n,h,t) = A234953(n+1) for n>0.
Sum_{h=0..n} Sum_{t=0..n-h} (h+1)*(n+1) * F(n,h,t) = A001854(n+1) for n>0.
Sum_{t=0..n-1} F(n,1,t) = A235596(n+1).
F(2n,n,n) = A126804(n) for n>0.
F(n,0,n) = 1 = A000012(n).
F(n,1,1) = n = A001477(n) for n>1.
F(n,n-1,1) = n! = A000142(n) for n>0.
F(n,1,n-1) = A002378(n-1) for n>0.
F(n,2,1) = A000551(n).
F(n,3,1) = A000552(n).
F(n,4,1) = A000553(n).
F(n,1,2) = A001788(n-1) for n>2.
F(n,0,0) = A000007(n).

cp's OEIS Frontend

A276289 Expansion of x*(1 + x)/(1 - 2*x)^3.

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A374497 Expansion of 1/(1 - 4*x - 4*x^2)^(3/2).

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A058395 Square array read by antidiagonals. Based on triangular numbers (A000217) with each term being the sum of 2 consecutive terms in the previous row.

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A059299 Triangle of idempotent numbers (version 3), T(n, k) = binomial(n, k) * (n - k)^k.

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A129532 3n(n-1)4^(n-2).

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A130812 If X_1,...,X_n is a partition of a 2n-set X into 2-blocks then a(n) is equal to the number of 6-subsets of X containing none of X_i, (i=1,...n).

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A158920 Binomial transform of A008805 (triangular numbers with repeats).

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A276289 Expansion of x(1 + x)/(1 - 2x)^3.

A374497 Expansion of 1/(1 - 4x - 4x^2)^(3/2).