A002699 -id:A002699 - cp's OEIS frontend

A002697 a(n) = n*4^(n-1).

0, 1, 8, 48, 256, 1280, 6144, 28672, 131072, 589824, 2621440, 11534336, 50331648, 218103808, 939524096, 4026531840, 17179869184, 73014444032, 309237645312, 1305670057984, 5497558138880, 23089744183296

Offset: 0

N. J. A. Sloane

Coefficient of x^(2n-2) in Chebyshev polynomial T(2n) is -a(n).
Let M_n be the n X n matrix m_(i,j) = 1 + 2*abs(i-j); then det(M_n) = (-1)^(n-1)*a(n-1). - Benoit Cloitre, May 28 2002
Number of subsequences 00 in all words of length n+1 on the alphabet {0,1,2,3}. Example: a(2)=8 because we have 000,001,002,003,100,200,300 (the other 57=A125145(3) words of length 3 have no subsequences 00). a(n) = Sum_{k=0..n} k*A128235(n+1, k). - Emeric Deutsch, Feb 27 2007
Let P(A) be the power set of an n-element set A. Then a(n) = the sum of the size of the symmetric difference of x and y for every subset {x,y} of P(A). - Ross La Haye, Dec 30 2007 (See the comment from Bernard Schott below.)
Let P(A) be the power set of an n-element set A and B be the Cartesian product of P(A) with itself. Then remove (y,x) from B when (x,y) is in B and x != y and call this R35. Then a(n) = the sum of the size of the symmetric difference of x and y for every (x,y) of R35. [proposed edit of comment just above; by Ross La Haye]
The numbers in this sequence are the Wiener indices of the graphs of n-cubes (Boolean hypercubes). For example, the 3-cube is the graph of the standard cube whose Wiener index is 48. - K.V.Iyer, Feb 26 2009
From Gary W. Adamson, Sep 06 2009: (Start)
Starting (1, 8, 48, ...) = 4th binomial transform of [1, 4, 0, 0, 0, ...].
Equals the sum of terms in 2^n X 2^n semi-magic square arrays in which each row and column is composed of a binomial frequency of terms in the set (1, 3, 5, 7, ...).
The first few such arrays = [1] [1,3; 3,1]; /Q.
  [1, 3, 5, 3;
   3, 1, 3, 5;
   5, 3, 1, 3;
   3, 5, 3, 1]
(sum of terms = 48, with a binomial frequency of (1, 2, 1) as to (1, 3, 5) in each row and column)
  [1, 3, 5, 3, 5, 7, 5, 3;
   3, 1, 3, 5, 7, 5, 3, 5;
   5, 3, 1, 3, 5, 3, 5, 7;
   3, 5, 3, 1, 3, 5, 7, 5;
   5, 7, 5, 3, 1, 3, 5, 3;
   7, 5, 3, 5, 3, 1, 3, 5;
   5, 3, 5, 7, 5, 3, 1, 3;
   3, 5, 7, 5, 3, 5, 3, 1]
(sum of terms = 256, with each row and column composed of one 1, three 3's, three 5's, and one 7)
... (End)
Let P(A) be the power set of an n-element set A and B be the Cartesian product of P(A) with itself. Then a(n) = the sum of the size of the intersection of x and y for every (x,y) of B. - Ross La Haye, Jan 05 2013
Following the last comment of Ross, A002699 is the similar sequence when "intersection" is replaced by "symmetric difference" and A212698 is the similar sequence when "intersection" is replaced by "union". - Bernard Schott, Jan 04 2013
Also, following the first comment of Ross, A082134 is the similar sequence when "symmetric difference" is replaced by "intersection" and A133224 is the similar sequence when "symmetric difference" is replaced by "union". - Bernard Schott, Jan 15 2013
Let [n] denote the set {1,2,3,...,n} and denote an n-permutation of the elements of [n] by p = p(1)p(2)p(3)...p(n), where p(i) is the i-th entry in the linear order given by p. Then (p(i),p(j)) is an inversion of p if i < j but p(i) > p(j). Denote the number of inversions of p by inv(p) and call a 2n-permutation p = p(1)p(2)...p(2n) 2-ordered if p(1) < p(3) < ... < p(2n-1) and p(2) < p(4) < ... < p(2n). Then Sum(inv(p)) = n*4^(n-1), where the sum is taken over all 2-ordered 2n-permutations of p. See Bona reference below. - Ross La Haye, Jan 21 2014
Sum over all peaks of Dyck paths of semilength n of the product of the x and y coordinates. - Alois P. Heinz, May 29 2015
Sum of the number of all edges over all j-dimensional subcubes of the boolean hypercube graph of dimension n, Q_n, for all j, so a(n) = Sum_{j=1..n} binomial(n,j)*2^(n-j) * j*2^(j-1). - Constantinos Kourouzides, Mar 24 2024

			From _Bernard Schott_, Jan 04 2013: (Start)
See the comment about intersection of X and Y.
If A={b,c}, then in P(A) we have:
{b}Inter{b}={b},
{b}Inter{b,c}={b},
{c}Inter{c}={c},
{c}Inter{b,c}={c},
{b,c}Inter{b}={b},
{b,c}Inter{c}={c},
{b,c}Inter{b,c}={b,c}
and : #{b}+ #{b}+ #{c}+ #{c}+ #{b}+ #{c}+ #{b,c} = 8 = 2*4^(2-1) = a(2).
The other intersections are empty.
(End)

Miklos Bona, Combinatorics of Permutations, Chapman and Hall/CRC, 2004, pp. 1, 43, 64.
C. Lanczos, Applied Analysis. Prentice-Hall, Englewood Cliffs, NJ, 1956, p. 516.
N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Danny Rorabaugh, Table of n, a(n) for n = 0..1000
F. Ellermann, Illustration of binomial transforms
Samuele Giraudo, Pluriassociative algebras I: The pluriassociative operad, arXiv:1603.01040 [math.CO], 2016.
INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 414
Milan Janjić, Pascal Matrices and Restricted Words, J. Int. Seq., Vol. 21 (2018), Article 18.5.2.
Constantinos Kourouzides, A double counting argument on the hypercube graph
Ross La Haye, Binary Relations on the Power Set of an n-Element Set, Journal of Integer Sequences, Vol. 12 (2009), Article 09.2.6.
C. Lanczos, Applied Analysis (Annotated scans of selected pages)
Aleksandar Petojević, A Note about the Pochhammer Symbol, Mathematica Moravica, Vol. 12-1 (2008), 37-42.
Simon Plouffe, Approximations de séries génératrices et quelques conjectures, Dissertation, Université du Québec à Montréal, 1992; arXiv:0911.4975 [math.NT], 2009.
Simon Plouffe, 1031 Generating Functions, Appendix to Thesis, Montreal, 1992
Eric Weisstein's World of Mathematics, Hypercube Graph
Eric Weisstein's World of Mathematics, Wiener Index
Index entries for linear recurrences with constant coefficients, signature (8,-16).

Cf. A000051, A000302, A000984, A001792, A002457, A002699, A027656, A038231, A082134, A083672, A125145, A128235, A133224, A212698.

A002697:=1/(4*z-1)**2; # conjectured by Simon Plouffe in his 1992 dissertation
A002697:=n->n*4^(n-1): seq(A002697(n), n=0..30); # Wesley Ivan Hurt, Mar 30 2014

Table[n 4^(n - 1), {n, 0, 30}] (* Harvey P. Dale, Jan 18 2012 *)
LinearRecurrence[{8, -16}, {0, 1}, 30] (* Harvey P. Dale, Jan 18 2012 *)
CoefficientList[Series[x/(1 - 4 x)^2, {x, 0, 20}], x] (* Eric W. Weisstein, Sep 08 2017 *)

PARI
```
a(n)=if(n<0,0,n*4^(n-1))
```

[n*4^(n-1) for n in range(22)] # Danny Rorabaugh, Mar 27 2015

a(n) = n*4^(n-1).
G.f.: x/(1-4x)^2. a(n+1) is the convolution of powers of 4 (A000302). - Wolfdieter Lang, May 16 2003
Third binomial transform of n. E.g.f.: x*exp(4x). - Paul Barry, Jul 22 2003
a(n) = Sum_{k=0..n} k*binomial(2*n, 2*k). - Benoit Cloitre, Jul 30 2003
For n>=0, a(n+1) = Sum_{i+j+k+l=n} binomial(2i, i)*binomial(2j, j)*binomial(2k, k)*binomial(2l, l). - Philippe Deléham, Jan 22 2004
a(n) = Sum_{k=0..n} 4^(n-k)*binomial(n-k+1, k)*binomial(1, (k+1)/2)*(1-(-1)^k)/2. - Paul Barry, Oct 15 2004
Sum_{n>0} 1/a(n) = 8*log(2) - 4*log(3). - Jaume Oliver Lafont, Sep 11 2009
a(0) = 0, a(n) = 4*a(n-1) + 4^(n-1). - Vincenzo Librandi, Dec 31 2010
a(n+1) is the convolution of A000984 with A002457. - Rui Duarte, Oct 08 2011
a(0) = 0, a(1) = 1, a(n) = 8*a(n-1) - 16*a(n-2). - Harvey P. Dale, Jan 18 2012
a(n) = A002699(n)/2 = A212698(n)/3. - Bernard Schott, Jan 04 2013
G.f.: W(0)*x/2 , where W(k) = 1 + 1/( 1 - 4*x*(k+2)/( 4*x*(k+2) + (k+1)/W(k+1) )); (continued fraction). - Sergei N. Gladkovskii, Aug 19 2013
Sum_{n>=1} (-1)^(n+1)/a(n) = 4*log(5/4). - Amiram Eldar, Oct 28 2020
a(n) = (1/2)*Sum_{k=0..n} k*binomial(2*n, k). Compare this with the formula of Benoit Cloitre above. - Wolfdieter Lang, Nov 12 2021
a(n) = (-1)^(n-1)*det(M(n)) for n > 0, where M(n) is the n X n symmetric Toeplitz matrix whose first row consists of 1, 3, ..., 2*n-1. - Stefano Spezia, Aug 04 2022

A053125 Triangle of coefficients of Chebyshev's U(n,2*x-1) polynomials (exponents of x in decreasing order).

Original entry on oeis.org

1, 4, -2, 16, -16, 3, 64, -96, 40, -4, 256, -512, 336, -80, 5, 1024, -2560, 2304, -896, 140, -6, 4096, -12288, 14080, -7680, 2016, -224, 7, 16384, -57344, 79872, -56320, 21120, -4032, 336, -8, 65536, -262144, 430080, -372736, 183040, -50688, 7392, -480, 9, 262144, -1179648, 2228224, -2293760, 1397760

Offset: 0

Wolfdieter Lang

A000302 (powers of 4), A002699, A002700 unsigned column sequences for m=0..2.
G.f. for row polynomials U(n,2*x-1) and row sums same as for A053124.
With offset 1 this is also the coefficient triangle of 2* U(2*n-1,x) expanded in decreasing powers of x. W. Lang, Mar 07 2007.

			{1}; {4,-2}; {16,-16,3}; {64,-96,40,-4}; {256,-512,336,-80,5};... E.g. fourth row (n=3) corresponds to polynomial U^{*}(3,m)=U(3,2*x-1)= 64*x^3-96*x^2+40*x-4.

C. Lanczos, Applied Analysis. Prentice-Hall, Englewood Cliffs, NJ, 1956, p. 518.
Theodore J. Rivlin, Chebyshev polynomials: from approximation theory to algebra and number theory, 2. ed., Wiley, New York, 1990.

T. D. Noe, Rows n=0..50 of triangle, flattened
W. Lang, First rows and related triangles .
Eric Weisstein's World of Mathematics, Chebyshev Polynomial of the Second Kind
Index entries for sequences related to Chebyshev polynomials.

Cf. A053124, A053123.

Reverse /@ CoefficientList[Table[ChebyshevU[n, 2 x - 1], {n, 0, 10}], x] // Flatten (* Eric W. Weisstein, Apr 04 2018 *)
Reverse /@ CoefficientList[ChebyshevU[Range[0, 10], 2 x - 1], x] // Flatten (* Eric W. Weisstein, Apr 04 2018 *)

a(n, m) = A053124(n, n-m)= (4^(n-m))*A053123(n, m)= (4^(n-m))*((-1)^m)*binomial(2*n+1-m, m) if n >= m, else 0.

a(n, m) := -2*a(n-1, m-1)+4*a(n-1, m)-a(n-2, m-2), a(-2, m) := 0=: a(n, -2), a(-1, m) := 0=: a(n, -1), a(0, 0)=1, a(n, m)=0 if n

G.f. for m-th column (signed triangle): ((-x)^m)*Po(m+1, 4*x)/(1-4*x)^(m+1), with Po(k, x) := sum('binomial(k, 2*j+1)*x^j', 'j'=0..floor(k/2)).

A212698 Main transitions in systems of n particles with spin 3/2.

Original entry on oeis.org

3, 24, 144, 768, 3840, 18432, 86016, 393216, 1769472, 7864320, 34603008, 150994944, 654311424, 2818572288, 12079595520, 51539607552, 219043332096, 927712935936, 3917010173952, 16492674416640, 69269232549888, 290271069732864, 1213860837064704, 5066549580791808

Offset: 1

Stanislav Sykora, May 25 2012

Please refer to the general explanation in A212697. This particular sequence is obtained for base b=4, corresponding to spin S = (b-1)/2 = 3/2.
Let P(A) be the power set of an n-element set A and let B be the Cartesian product of P(A) with itself. Then a(n) = the sum of the size of the union of x and y for every (x,y) in B. [See Relation (28): U(n) in document of Ross La Haye in reference.] - Bernard Schott, Jan 04 2013
A002697 is the analogous sequence if "union" is replaced by "intersection" and A002699 is the analogous sequence if "union" is replaced by "symmetric difference". Here, X union Y and Y union X are considered as two distinct Cartesian products, if we want to consider that X Union Y = Y Union X are the same Cartesian product, see A133224. - Bernard Schott Jan 11 2013

Stanislav Sykora, Table of n, a(n) for n = 1..100
Ross La Haye, Binary Relations on the Power Set of an n-Element Set, Journal of Integer Sequences, Vol. 12 (2009), Article 09.2.6.
Stanislav Sýkora, Magnetic Resonance on OEIS, Stan's NMR Blog (Dec 31, 2014), Retrieved Nov 12, 2019.
Index entries for linear recurrences with constant coefficients, signature (8,-16).

Cf. A001787, A212697, A212699, A212700, A212701, A212702, A212703, A212704 (for b = 2, 3, 5, 6, 7, 8, 9, 10).

Cf. A000302, A002697, A002699, A008585, A133224.

[3*n*4^(n-1): n in [1..30]]; // Vincenzo Librandi, Nov 29 2015

Table[Sum[Binomial[n,i] i 3^i,{i,0,n}],{n,1,21}] (* Geoffrey Critzer, Aug 08 2013 *)

mtrans(n, b) = n*(b-1)*b^(n-1);
for (n=1, 100, write("b212698.txt", n, " ", mtrans(n, 4)))

a(n) = n*(b-1)*b^(n-1). For this sequence, set b=4.
a(n) = 3*n*4^(n-1).
a(n) = 3*A002697(n).
From Geoffrey Critzer, Aug 08 2013: (Start)
a(n) = Sum_{i>=0} binomial(n,i)*i*3^i.
E.g.f.: 3*x*exp(4*x). (End)
G.f.: 3*x/(4*x-1)^2. - Colin Barker, Nov 03 2014
From Elmo R. Oliveira, May 24 2025: (Start)
a(n) = 8*a(n-1) - 16*a(n-2) for n > 2.
a(n) = A008585(n)*A000302(n-1). (End)

A002700 Coefficients of Chebyshev polynomials: n(2n+1) * 4^(n-1).

Original entry on oeis.org

3, 40, 336, 2304, 14080, 79872, 430080, 2228224, 11206656, 55050240, 265289728, 1258291200, 5888802816, 27246198784, 124822487040, 566935683072, 2555505541120, 11441792876544, 50921132261376, 225399883694080, 992858999881728, 4354066045992960

Offset: 1

N. J. A. Sloane

Cornelius Lanczos, Applied Analysis. Prentice-Hall, Englewood Cliffs, NJ, 1956, p. 518.
N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Colin Barker, Table of n, a(n) for n = 1..1000
Cornelius Lanczos, Applied Analysis. (Annotated scans of selected pages)
Simon Plouffe, Approximations de séries génératrices et quelques conjectures, Dissertation, Université du Québec à Montréal, 1992; arXiv:0911.4975 [math.NT], 2009.
Simon Plouffe, 1031 Generating Functions, Appendix to Thesis, Montreal, 1992.
Index entries for linear recurrences with constant coefficients, signature (12,-48,64).
Index entries for sequences related to Chebyshev polynomials.

Cf. A002699.

List([1..30], n-> 4^(n-1)*n*(2*n+1)); # G. C. Greubel, Jul 23 2019

[4^(n-1)*n*(2*n+1): n in [1..30]]; // G. C. Greubel, Jul 23 2019

A002700:=-(3+4*z)/(4*z-1)**3; # Simon Plouffe in his 1992 dissertation.

Table[n*(2*n+1)*2^(2*n-2),{n,1,30}] (* Vaclav Kotesovec, Jun 03 2014 *)
LinearRecurrence[{12,-48,64},{3,40,336},30] (* Harvey P. Dale, May 17 2018 *)

Vec(-x*(4*x+3)/(4*x-1)^3 + O(x^30)) \\ Colin Barker, Jun 15 2015

[4^(n-1)*n*(2*n+1) for n in (1..30)] # G. C. Greubel, Jul 23 2019

a(n) = 12*a(n-1) - 48*a(n-2) + 64*a(n-3). - Colin Barker, Jun 15 2015
a(n) = 1/2*Sum_{k = 0..2*n} k^2*binomial(2*n,k). Cf. A002699. - Peter Bala, Apr 09 2017
From Amiram Eldar, Feb 17 2023: (Start)
Sum_{n>=1} 1/a(n) = 8 + 8*log(2) - 12*log(3).
Sum_{n>=1} (-1)^(n+1)/a(n) = 16*arctan(1/2) + 4*log(5/4) - 8. (End)

A254632 Triangle read by rows, T(n, k) = 4^n*[x^k]hypergeometric([3/2, -n], [3], -x), n>=0, 0<=k<=n.

Original entry on oeis.org

1, 4, 2, 16, 16, 5, 64, 96, 60, 14, 256, 512, 480, 224, 42, 1024, 2560, 3200, 2240, 840, 132, 4096, 12288, 19200, 17920, 10080, 3168, 429, 16384, 57344, 107520, 125440, 94080, 44352, 12012, 1430, 65536, 262144, 573440, 802816, 752640, 473088, 192192, 45760, 4862

Offset: 0

Peter Luschny, Feb 03 2015

			[   1]
[   4,     2]
[  16,    16,     5]
[  64,    96,    60,    14]
[ 256,   512,   480,   224,    42]
[1024,  2560,  3200,  2240,   840,  132]
[4096, 12288, 19200, 17920, 10080, 3168, 429]

Cf. A108198 (Peter Bala), A000302, A000108, A025230, A002699, A128650, A254633.

h := n -> simplify(hypergeom([3/2, -n], [3], -x)):
seq(print(seq(4^n*coeff(h(n), x, k), k=0..n)), n=0..9);

T[n_, k_] := 4^(n-k) Binomial[n, k] CatalanNumber[k+1];
Table[T[n, k], {n, 0, 8}, {k, 0, n}] (* Jean-François Alcover, Jun 28 2019 *)

A254632 = lambda n,k: (4)^(n-k)*binomial(n,k)*catalan_number(k+1)
for n in range(7): [A254632(n,k) for k in (0..n)]

T(n,0) = A000302(n).
T(n,n) = A000108(n+1).
T(n,1) = A002699(n) for n>=1.
T(n,n-1) = A128650(n+2) for n>=1.
T(2*n,n) = A254633(n).
T(n,k) = 4^(n-k)*C(n,k)*Catalan(k+1).
sum(k=0..n, T(n,k)) = A025230(n+2).

A328000 a(n) = Sum_{k=0..n}(k!(n - k)!)/(floor(k/2)!floor((n - k)/2)!)^2.

Original entry on oeis.org

1, 2, 5, 16, 28, 96, 160, 512, 896, 2560, 4864, 12288, 25600, 57344, 131072, 262144, 655360, 1179648, 3211264, 5242880, 15466496, 23068672, 73400320, 100663296, 343932928, 436207616, 1593835520, 1879048192, 7314866176, 8053063680, 33285996544, 34359738368

Offset: 0

Peter Luschny, Oct 01 2019

Colin Barker, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (0,12,0,-48,0,64).

Cf. A327999, A328001.

Cf. A056040, A002699, A001477, A081908, A145018.

[IsOdd(n) select 2^(n - 1)*(n + 1) else 2^(n - 5)*(n*(n + 2) + 32):n in [0..30]]; // Marius A. Burtea, Feb 05 2020

swing := n -> n!/iquo(n,2)!^2: a := n -> add(swing(k)*swing(n-k), k=0..n):
seq(`if`(irem(n, 2) = 0, 2 + n*(n + 2)/16, n + 1)*2^(n - 1), n=0..31);

A328000List[len_] := CoefficientList[Series[(4 x^2 - x - 1)^2 / (1 - 4 x^2)^3 , {x, 0, len}], x]; A328000List[31]
LinearRecurrence[{0,12,0,-48,0,64},{1,2,5,16,28,96},40] (* Harvey P. Dale, Jun 19 2022 *)

x='x + O('x^32);
Vec(serlaplace(((3*x + 8)*sinh(2*x) + (2*x^2 + 16*(x + 1))*cosh(2*x))/16))

Vec((1 + x - 4*x^2)^2 / ((1 - 2*x)^3*(1 + 2*x)^3) + O(x^30)) \\ Colin Barker, Feb 05 2020

a(n) = Sum_{k=0..n} s(k)*s(n-k) where s(n) = A056040(n).
a(n) = [x^n] (4*x^2 - x - 1)^2 / (1 - 4*x^2)^3.
a(n) = 2^(n - 5)*(n*(n + 2) + 32) if n even else 2^(n - 1)*(n + 1).
a(2*n) = A327999(n).
a(2*n-1) = A002699(n), (with a(-1) = 0).
a(2^n-1) = 2^(2^n - 2 + n) for n >= 1.
2*a(2*n)/2^n = A081908(n+1).
4*a(2*n)/4^n = A145018(n+1).
2*a(2*n-1)/4^n = A001477(n).
From Stefano Spezia, Oct 19 2019: (Start)
a(n) = n! [x^n] (1/32)*exp(-2*x)*(8 + exp(4*x)*(8 + x)*(3 + 2*x) + x*(13 + 2*x)).
a(n) = 12*a(n-2) - 48*a(n-4) + 64*a(n-6) for n > 5. (End)

A303602 a(n) = Sum_{k = 0..n} kbinomial(2n+1, k).

Original entry on oeis.org

0, 3, 25, 154, 837, 4246, 20618, 97140, 447661, 2028478, 9070110, 40122028, 175913250, 765561564, 3310623412, 14238676712, 60949133949, 259809601870, 1103420316566, 4670886541308, 19714134528598, 82985455688276, 348481959315660, 1460179866076504, 6106070639175122

Offset: 0

Bruno Berselli, May 09 2018

Second bisection of A185251; the first bisection is A002699.

The terms are not congruent to 5 (mod 6).

Cf. A000346, A002699, A005408, A047226, A164991, A185251.

seq(add(k*binomial(2*n+1,k),k=0..n),n=0..24); # Paolo P. Lava, May 10 2018

Table[Sum[k Binomial[2 n + 1, k], {k, 0, n}], {n, 0, 30}]
CoefficientList[Series[(1 + 4*x - Sqrt[1 - 4*x]) / (2*(1 - 4*x)^2), {x, 0, 25}], x] (* Vaclav Kotesovec, May 10 2018 *)

a(n)=(2*n+1)*(4^n-binomial(2*n,n))/2 \\ Charles R Greathouse IV, Oct 23 2023

[(2*n+1)*(4^n-binomial(2*n,n))/2 for n in (0..30)]

E.g.f.: ((1 + 8*x)*exp(2*x) - (1 + 4*x)*I_0(2*x) - 4*x*I_1(2*x))*exp(2*x)/2, where I_m(.) is the modified Bessel function of the first kind.
From Vaclav Kotesovec, May 10 2018: (Start)
G.f.: (1 + 4*x - sqrt(1 - 4*x)) / (2*(1 - 4*x)^2).
D-finite with recurrence: n*(2*n-1)*a(n) = 2*(2*n+1)*(4*n-3)*a(n-1) - 8*(2*n-1)*(2*n+1)*a(n-2). (End)
a(n) = (2*n + 1)*(4^n - binomial(2*n, n))/2.
a(n+1) - 4*a(n) = A164991(2*n+3).

A375853 Triangle read by rows: T(n, k) = k(n - k)binomial(2n+2, 2k+1)/(4*n + 2) for 1 <= k <= n-1.

Original entry on oeis.org

2, 8, 8, 20, 56, 20, 40, 216, 216, 40, 70, 616, 1188, 616, 70, 112, 1456, 4576, 4576, 1456, 112, 168, 3024, 14040, 22880, 14040, 3024, 168, 240, 5712, 36720, 88400, 88400, 36720, 5712, 240, 330, 10032, 85272, 284240, 419900, 284240, 85272, 10032, 330

Offset: 2

Mingjian Ding, Aug 31 2024

The T(n, k) are the coefficients of the minuscule polynomials of type A. They are the Wiener index of a minuscule lattice of type A, i.e., the Hasse diagram of the poset of order ideals in a k X (n - k) rectangle.

			Triangle begins:
  n\k  1    2     3    4   5
  2:   2;
  3:   8,   8;
  4:  20,  56,   20;
  5:  40, 216,  216,  40;
  6:  70, 616, 1188, 616, 70;
 ...

Rebecca Bourn and Jeb F. Willenbring, Expected value of the one-dimensional earth mover's distance, Algebr. Stat. 11 (2020), no. 1, 53-78.
Rebecca Bourn and William Q. Erickson, Proof of a conjecture of Bourn and Willenbring concerning a family of palindromic polynomials, arXiv:2307.02652 [math.CO], 2023.
Colin Defant, Valentin Féray, Philippe Nadeau, and Nathan Williams, Wiener indices of minuscule lattices, Electron. J. Combin. 31 (2024), no.1, Paper No. 1.41, 23 pp.
Ming-Jian Ding and Jiang Zeng, Some new results on minuscule polynomial of type A, arXiv:2308.16782 [math.CO], 2023.
Eric Weisstein's World of Mathematics, Wiener Index.

Column 1 and main diagonal are A007290(n+1).
Row sums are A002699(n-1).
Half the sums of the gamma coefficients are A376072(n).

Trow := n -> seq(1/(4*n+2)*k*(n-k)*binomial(2*n+2, 2*k+1), k = 1..n-1):
for n from 2 to 10 do Trow(n) od;
# Alternatively, using the generating function of the row polynomials:
rgf := (n, x) -> ((sqrt(x) - 1)^(2*n)*(2*n*sqrt(x) + x + 1) - (sqrt(x) + 1)^(2*n)*(-2*n*sqrt(x) + x + 1))/(16*sqrt(x)):
T := (n, k) -> coeff(expand(rgf(n, x)), x, k):
seq(print(seq(T(n, k), k = 1..n - 1)), n = 2..8): # Peter Luschny, Sep 22 2024

Flatten@Table[k*(n - k)*Binomial[2*n + 2, 2*k + 1]/(4*n + 2), {n, 2, 10}, {k, n - 1}] (* Zhining Yang, Sep 18 2024 *)

T(n,k) = k*(n-k)*binomial(2*n+2,2*k+1)/(4*n+2) \\ Andrew Howroyd, Sep 01 2024

Sum_{k>=0} T(n, k) = A002699(n-1) (conjectured by Bourn and Erickson).
G.f.: T_n(x) = Sum_{k>=0} T(n, k)*x^k = (1 - x)^{2*n}*Sum_{k>=0}Sum_{alpha, beta} EMD_k(alpha, beta)*x^k, where EMD_k is the Earth Mover's Distance on (alpha, beta), and alpha, beta are the elements of composition of k into n parts.
T_n(x^2) = (n + 1)/8*((1 + x)^(2*n) + (1 - x)^(2*n)) - 1/(16*x)*((1 + x)^(2*n + 2) - (1 - x)^(2*n + 2)). (Proposition 3.1, arXiv:2308.16782)

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cp's OEIS Frontend

A002697 a(n) = n*4^(n-1).

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A053125 Triangle of coefficients of Chebyshev's U(n,2*x-1) polynomials (exponents of x in decreasing order).

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A212698 Main transitions in systems of n particles with spin 3/2.

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A002700 Coefficients of Chebyshev polynomials: n*(2*n+1) * 4^(n-1).

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A254632 Triangle read by rows, T(n, k) = 4^n*[x^k]hypergeometric([3/2, -n], [3], -x), n>=0, 0<=k<=n.

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A328000 a(n) = Sum_{k=0..n}(k!*(n - k)!)/(floor(k/2)!*floor((n - k)/2)!)^2.

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A303602 a(n) = Sum_{k = 0..n} k*binomial(2*n+1, k).

A002700 Coefficients of Chebyshev polynomials: n(2n+1) * 4^(n-1).

A328000 a(n) = Sum_{k=0..n}(k!(n - k)!)/(floor(k/2)!floor((n - k)/2)!)^2.

A303602 a(n) = Sum_{k = 0..n} kbinomial(2n+1, k).

A375853 Triangle read by rows: T(n, k) = k(n - k)binomial(2n+2, 2k+1)/(4*n + 2) for 1 <= k <= n-1.