A080951 -id:A080951 - cp's OEIS frontend

A057711 a(0)=0, a(1)=1, a(n) = n*2^(n-2) for n >= 2.

0, 1, 2, 6, 16, 40, 96, 224, 512, 1152, 2560, 5632, 12288, 26624, 57344, 122880, 262144, 557056, 1179648, 2490368, 5242880, 11010048, 23068672, 48234496, 100663296, 209715200, 436207616, 905969664, 1879048192, 3892314112, 8053063680

Offset: 0

Bernhard Wolf (wolf(AT)cs.tu-berlin.de), Oct 24 2000

Number of states in the planning domain FERRY, when n-3 cars are at one of two shores while the (n-2)nd car may be on the ferry or at one of the shores.
If the ferry could board any number of cars (instead of only one), the number of states would form the Pisot sequence P(2,6) (A008776). In addition, if k shores existed, the sequence would form the Pisot sequence P(k,k(k+1)). This corresponds to the BRIEFCASE planning domain.
a(i) is the number of occurrences of the number 1 in all palindromic compositions of n = 2*(i+1). - Silvia Heubach (sheubac(AT)calstatela.edu), Jan 10 2003. E.g., there are 5 palindromic compositions of 6, namely 111111 11211 2112 1221 141, containing a total of 16 1's.
Number of occurrences of 00's in all circular binary words of length n. Example: a(3)=6 because in the circular binary words 000, 001, 010, 011, 100, 101, 110 and 111 we have a total of 3+1+1+0+1+0+0+0=6 occurrences of 00. a(n) = Sum_{k=0..n} k*A119458(n,k). - Emeric Deutsch, May 20 2006
a(n) is the number of permutations on [n] for which the entries of each left factor form a circular subinterval of [n]. A subset I of [n] forms a circular subinterval of [n] if it is an ordinary interval [a,b] or has the form [1,a]-union-[b,n] for 1 <= a < b <= n. For example, (5,4,2) is a left factor of the permutation (5,4,2,1,3) which does not form a circular subinterval of [5] and a(4)=16 counts all 24 permutations of [4] except the eight whose first two entries are 1,3 (in either order) or 2,4. - David Callan, Mar 30 2007
a(n) is the total number of runs in all Boolean (n-1)-strings. For example, the 8 Boolean 3-strings, 000, 001, 010, 011, 100, 101, 110, 111 have 1, 2, 3, 2, 2, 3, 2, 1 runs respectively. - David Callan, Jul 22 2008
From Gary W. Adamson, Jul 31 2010: (Start)
Starting with "1" = (1, 2, 4, 8, ...) convolved with (1, 0, 2, 4, 8, ...).
Example: a(6) = 96 = (32, 16, 8, 4, 2, 1) dot (1, 0, 2, 4, 8, 16) = (32 + 0 + 16 + 16 + 16, + 16) = 32 + 4*16 (End)
An elephant sequence, see A175654. For the corner squares 24 A[5] vectors, with decimal values between 27 and 432, lead to this sequence (without the leading 0). For the central square these vectors lead to the companion sequence A087447 (without the first leading 1). - Johannes W. Meijer, Aug 15 2010
Starting with 1 = (1, 1, 2, 4, 8, 16, ...) convolved with (1, 1, 3, 7, 15, 31, ...). - Gary W. Adamson, Oct 26 2010
a(n) is the number of ways to draw simple polygonal chains for n vertices lying on a circle. - Anton Zakharov, Dec 31 2016
Also the number of edges, maximal cliques, and maximum cliques in the n-folded cube graph for n > 3. - Eric W. Weisstein, Dec 01 2017 and Mar 21 2018
Number of pairs of compositions of n corresponding to a seaweed algebra of index n-2 for n > 2. - Nick Mayers, Jun 25 2018
Starting with 1, 2, 6, 16, ..., number of permutations of length n>0 avoiding the partially ordered pattern (POP) {1>2, 1>3} of length 4. That is, number of length n permutations having no subsequences of length 4 in which the first element is larger than the second and third elements. - Sergey Kitaev, Dec 08 2020

			a(1)=6 because the palindromic compositions of n=4 are 4, 1+2+1, 1+1+1+1 and 2+2 and they contain 6 ones. - Silvia Heubach (sheubac(AT)calstatela.edu), Jan 10 2003

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
O. Aichholzer, A. Asinowski, and T. Miltzow, Disjoint compatibility graph of non-crossing matchings of points in convex position, arXiv preprint arXiv:1403.5546 [math.CO], 2014.
A. Burstein, S. Kitaev, and T. Mansour, Partially ordered patterns and their combinatorial interpretations, PU. M. A. Vol. 19 (2008), No. 2-3, pp. 27-38.
P. Chinn, R. Grimaldi and S. Heubach, The frequency of summands of a particular size in Palindromic Compositions, Ars Combin. 69 (2003), 65-78.
Vincent Coll et al., Meander graphs and Frobenius seaweed Lie algebras II, Journal of Generalized Lie Theory and Applications 9.1 (2015).
Vladimir Dergachev and Alexandre Kirillov, Index of Lie algebras of seaweed type, J. Lie Theory 10.2 (2000): 331-343.
Alice L. L. Gao and Sergey Kitaev, On partially ordered patterns of length 4 and 5 in permutations, arXiv:1903.08946 [math.CO], 2019.
Alice L. L. Gao and Sergey Kitaev, On partially ordered patterns of length 4 and 5 in permutations, The Electronic Journal of Combinatorics 26(3) (2019), P3.26.
M. Ghallab et al., FERRY domain
M. Ghallab, A. Howe et al., PDDL - The Planning Domain Definition Language, Version 1.2, Technical Report CVC TR-98-003/DCS TR-1165. Yale Center for Computational Vision and Control, 1998.
Anna Khmelnitskaya, Gerard van der Laan, and Dolf Talmanm, The Number of Ways to Construct a Connected Graph: A Graph-Based Generalization of the Binomial Coefficients, J. Int. Seq. (2023) Art. 23.4.3. See p. 12.
Eric Weisstein's World of Mathematics, Folded Cube Graph
Eric Weisstein's World of Mathematics, Maximal Clique
Eric Weisstein's World of Mathematics, Maximum Clique
B. Wolf, Creating state sets
Index entries for linear recurrences with constant coefficients, signature (4,-4).

Cf. A082133, A082134, A082135, A082136.
Pisot sequence P(2, 6) (A008776), Pisot sequence P(k, k(k+1))
Cf. A119458.
Cf. A082140, A082141, A082138, A082139, A080951, A080929.

[Ceiling(n*2^(n-2)) : n in [0..40]]; // Vincenzo Librandi, Sep 22 2011

Join[{0, 1}, Table[n 2^(n - 2), {n, 2, 30}]] (* Eric W. Weisstein, Dec 01 2017 *)
Join[{0, 1}, LinearRecurrence[{4, -4}, {2, 6}, 20]] (* Eric W. Weisstein, Dec 01 2017 *)
CoefficientList[Series[x (1 - 2 x + 2 x^2)/(1 - 2 x)^2, {x, 0, 20}], x] (* Eric W. Weisstein, Dec 01 2017 *)

a(n)=ceil(n*2^(n-2)) \\ Charles R Greathouse IV, Oct 31 2011

x='x+O('x^50); concat(0, Vec(x*(1-2*x+2*x^2)/(1-2*x)^2)) \\ Altug Alkan, Nov 01 2015

a(n) = ceiling(n*2^(n-2)).
Binomial transform of (0, 1, 0, 3, 0, 5, 0, 7, ...).
From Paul Barry, Apr 06 2003: (Start)
a(0)=0, a(n) = n*(0^(n-1) + 2^(n-1))/2, n > 0.
a(n) = Sum_{k=0..n} binomial(n, 2k+1)*(2k+1).
E.g.f.: x*exp(x)*cosh(x). (End)
The sequence 1, 1, 6, 16, ... is the binomial transform of A016813 with interpolated zeros. - Paul Barry, Jul 25 2003
For n > 1, a(n) = Sum_{k=0..n} (k-n/2)^2 C(n, k). (n+1)*a(n) = A001788(n). - Mario Catalani (mario.catalani(AT)unito.it), Nov 26 2003
From Paul Barry, May 07 2004: (Start)
a(n) = n*2^(n-2) - Sum_{k=0..n} binomial(n, k)*k*(-1)^k.
G.f.: x*(1-2*x+2*x^2)/(1-2*x)^2. (End)
a(n+1) = ceiling(binomial(n+1,1)*2^(n-1)). - Zerinvary Lajos, Nov 01 2006
a(n+1) = Sum_{k=0..n} A196389(n,k)*2^k. - Philippe Deléham, Oct 31 2011
a(0)=0, a(1)=1, a(2)=2, a(3)=6, a(n+1) = 4*a(n)-4*a(n-1) for n >= 3. - Philippe Deléham, Feb 20 2013
a(n) = A002064(n-1) - A002064(n-2), for n >= 2. - Ivan N. Ianakiev, Dec 29 2013
From Amiram Eldar, Aug 05 2020: (Start)
Sum_{n>=1} 1/a(n) = 4*log(2) - 1.
Sum_{n>=1} (-1)^(n+1)/a(n) = 4*log(3/2) - 1. (End)

A082137 Square array of transforms of binomial coefficients, read by antidiagonals.

Original entry on oeis.org

1, 1, 1, 1, 2, 2, 1, 3, 6, 4, 1, 4, 12, 16, 8, 1, 5, 20, 40, 40, 16, 1, 6, 30, 80, 120, 96, 32, 1, 7, 42, 140, 280, 336, 224, 64, 1, 8, 56, 224, 560, 896, 896, 512, 128, 1, 9, 72, 336, 1008, 2016, 2688, 2304, 1152, 256, 1, 10, 90, 480, 1680, 4032, 6720, 7680, 5760, 2560, 512

Offset: 0

Paul Barry, Apr 06 2003

Rows are associated with the expansions of (x^k/k!)exp(x)cosh(x) (leading zeros dropped). Rows include A011782, A057711, A080929, A082138, A080951, A082139, A082140, A082141. Columns are of the form 2^(k-1)C(n+k, k). Diagonals include A069723, A082143, A082144, A082145, A069720.
T(n, k) is also the number of idempotent order-preserving and order-decreasing partial transformations (of an n-chain) of width k (width(alpha)= |Dom(alpha)|). - Abdullahi Umar, Oct 02 2008
Read as a triangle this is A119468 with rows reversed. A119468 has e.g.f. exp(z*x)/(1-tanh(x)). - Peter Luschny, Aug 01 2012
Read as a triangle this is a subtriangle of A198793. - Philippe Deléham, Nov 10 2013

			Rows begin
  1 1  2   4   8 ...
  1 2  6  16  40 ...
  1 3 12  40 120 ...
  1 4 20  80 280 ...
  1 5 30 140 560 ...
Read as a triangle, this begins:
  1
  1, 1
  1, 2,  2
  1, 3,  6,  4
  1, 4, 12, 16,   8
  1, 5, 20, 40,  40, 16
  1, 6, 30, 80, 120, 96, 32
  ... - _Philippe Deléham_, Nov 10 2013

G. C. Greubel, Table of n, a(n) for the first 50 rows, flattened
Laradji, A. and Umar, A. Combinatorial results for semigroups of order-preserving partial transformations, Journal of Algebra 278, (2004), 342-359.
Laradji, A. and Umar, A. Combinatorial results for semigroups of order-decreasing partial transformations, J. Integer Seq. 7 (2004), 04.3.8.

Cf. A119468, A007318, A134309.

# As a triangular array:
T := (n,k) -> 2^(k+0^k-1)*binomial(n,k):
for n from 0 to 9 do seq(T(n,k), k=0..n) od; # Peter Luschny, Nov 10 2017

rows = 11; t[n_, k_] := 2^(n-1)*(n+k)!/(n!*k!); t[0, ] = 1; tkn = Table[ t[n, k], {k, 0, rows}, {n, 0, rows}]; Flatten[ Table[ tkn[[ n-k+1, k ]], {n, 1, rows}, {k, 1, n}]] (* _Jean-François Alcover, Jan 20 2012 *)

def A082137_row(n) : # as a triangular array
    var('z')
    s = (exp(z*x)/(1-tanh(x))).series(x,n+2)
    t = factorial(n)*s.coefficient(x,n)
    return [t.coefficient(z,n-k) for k in (0..n)]
for n in (0..7) : print(A082137_row(n))  # Peter Luschny, Aug 01 2012

Square array defined by T(n, k)=(2^(n-1)+0^n/2)C(n + k, n)= Sum{k=0..n, C(n+k, k+j)C(k+j, k)(1+(-1)^j)/2 }.
As an infinite lower triangular matrix, equals A007318 * A134309. - Gary W. Adamson, Oct 19 2007
O.g.f. for array read as a triangle: (1-x*(1+t))/((1-x)*(1-x*(1+2*t))) = 1 + x*(1+t) + x^2*(1+2*t+2*t^2) + x^3*(1+3*t+6*t^2+4*t^3) + .... - Peter Bala, Apr 26 2012
For array read as a triangle: T(n,k) = 2*T(n-1,k) + 2*T(n-1,k-1) - T(n-2,k) -2*T(n-2,k-1), T(0,0) = T(1,0) = T(1,1) = 1, T(n,k) = 0 if k<0 or if k>n. - Philippe Deléham, Nov 10 2013

A080929 Sequence associated with a(n) = 2a(n-1) + k(k+2)*a(n-2).

Original entry on oeis.org

1, 3, 12, 40, 120, 336, 896, 2304, 5760, 14080, 33792, 79872, 186368, 430080, 983040, 2228224, 5013504, 11206656, 24903680, 55050240, 121110528, 265289728, 578813952, 1258291200, 2726297600, 5888802816, 12683575296, 27246198784

Offset: 0

Paul Barry, Feb 26 2003

The third column of number triangle A080928.

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (6,-12,8).

Essentially the same as A052482.

Cf. A082140, A082141, A082138, A082139, A080951, A080929, A057711.

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

[n eq 0 select 1 else (n+1)*(n+2)*2^(n-2): n in [0..30]]; // Vincenzo Librandi, Sep 22 2011

[seq (ceil(binomial(n+2,2)*2^(n-1)),n=0..30)]; # Zerinvary Lajos, Nov 01 2006

CoefficientList[Series[(1-x)(1-2x+4x^2)/(1-2x)^3, {x,0,30}], x] (* Michael De Vlieger, Sep 21 2017 *)
Join[{1}, LinearRecurrence[{6,-12,8}, {3,12,40}, 30]] (* G. C. Greubel, Jul 23 2019 *)

vector(30, n, n--; if(n==0,1, 2^(n-1)*binomial(n+2,2) )) \\ G. C. Greubel, Jul 23 2019

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

G.f.: (1-x)*(1-2*x+4*x^2)/(1-2*x)^3.
For n>0, a(n) = (n+1)*(n+2)*2^(n-2). - Ralf Stephan, Jan 16 2004
a(n) = Sum_{k=0..n} Sum_{i=0..n} (k+1)*binomial(n-1,i). - Wesley Ivan Hurt, Sep 20 2017
From Amiram Eldar, Jan 07 2022: (Start)
Sum_{n>=0} 1/a(n) = 7 - 8*log(2).
Sum_{n>=0} (-1)^n/a(n) = 24*log(3/2) - 9. (End)

A082139 A transform of binomial(n,5).

Original entry on oeis.org

1, 6, 42, 224, 1008, 4032, 14784, 50688, 164736, 512512, 1537536, 4472832, 12673024, 35094528, 95256576, 254017536, 666796032, 1725825024, 4410441728, 11142168576, 27855421440, 68975329280, 169303080960, 412216197120

Offset: 0

Paul Barry, Apr 06 2003

Sixth row of number array A082137. C(n,5) has e.g.f. (x^5/5!)exp(x). The transform averages the binomial and inverse binomial transforms.

			a(0) = (2^(-1) + 0^0/2)*binomial(5,0) = 2*(1/2) = 1 (use 0^0 = 1).

Vincenzo Librandi, Table of n, a(n) for n = 0..400
Index entries for linear recurrences with constant coefficients, signature (12,-60,160,-240,192,-64).

Cf. A080951, A080952, A082138.

Cf. A082140, A082141, A082138, A080951, A080929, A057711.

[(Ceiling(Binomial(n+5, 5)*2^(n-1))) : n in [0..30]]; // Vincenzo Librandi, Sep 22 2011

[seq (ceil(binomial(n+5,5)*2^(n-1)),n=0..23)]; # Zerinvary Lajos, Nov 01 2006

Drop[With[{nmax = 56}, CoefficientList[Series[x^5*Exp[x]*Cosh[x]/5!, {x, 0, nmax}], x]*Range[0, nmax]!], 5] (* or *) Join[{1}, Table[2^(n-1)* Binomial[n+5,n], {n,1,30}]] (* G. C. Greubel, Feb 05 2018 *)

my(x='x+O('x^30)); Vec(serlaplace(x^5*exp(x)*cosh(x)/5!)) \\ G. C. Greubel, Feb 05 2018

Equals 2 * A080952.
a(n) = (2^(n-1) + 0^n/2)*C(n+5, n).
a(n) = Sum_{j=0..n} C(n+5, j+5)*C(j+5, 5)*(1+(-1)^j)/2.
G.f.: (1 -6*x +30*x^2 -80*x^3 +120*x^4 -96*x^5 +32*x^6)/(1-2*x)^6.
E.g.f.: x^5*exp(x)*cosh(x)/5! (preceded by 5 zeros).
a(n) = ceiling(binomial(n+5,5)*2^(n-1)). - Zerinvary Lajos, Nov 01 2006
From Amiram Eldar, Jan 07 2022: (Start)
Sum_{n>=0} 1/a(n) = 20*log(2) - 38/3.
Sum_{n>=0} (-1)^n/a(n) = 1620*log(3/2) - 656. (End)

A082140 A transform of binomial(n,6).

Original entry on oeis.org

1, 7, 56, 336, 1680, 7392, 29568, 109824, 384384, 1281280, 4100096, 12673024, 38019072, 111132672, 317521920, 889061376, 2444918784, 6615662592, 17641766912, 46425702400, 120706826240, 310388981760, 790081044480

Offset: 0

Paul Barry, Apr 06 2003

Seventh row of number array A082137. C(n,6) has e.g.f. (x^6/6!)exp(x). The transform averages the binomial and inverse binomial transforms.

			a(0) = (2^(-1) + 0^0/2)*binomial(6,0) = 2*(1/2) = 1 (use 0^0 = 1).

G. C. Greubel, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (14,-84,280,-560,672,-448,128).

Cf. A082137, A082139, A080951.

For n>0, a(n) = 1/2 * A002409(n).

[(2^(n-1) + 0^n/2)*Binomial(n+6,n): n in [0..30]]; // G. C. Greubel, Feb 05 2018

[seq (ceil(binomial(n+6,6)*2^(n-1)),n=0..22)]; # Zerinvary Lajos, Nov 01 2006

Drop[With[{nmax = 56}, CoefficientList[Series[x^6*Exp[x]*Cosh[x]/6!, {x, 0, nmax}], x]*Range[0, nmax]!], 5] (* or *) Join[{1}, Table[2^(n-1)* Binomial[n+6,n], {n,1,30}]] (* G. C. Greubel, Feb 05 2018 *)
LinearRecurrence[{14,-84,280,-560,672,-448,128},{1,7,56,336,1680,7392,29568,109824},30] (* Harvey P. Dale, Jul 18 2023 *)

my(x='x+O('x^30)); Vec(serlaplace(x^6*exp(x)*cosh(x)/6!)) \\ G. C. Greubel, Feb 05 2018

a(n) = (2^(n-1) + 0^n/2)*C(n+6,n).
a(n) = Sum_{j=0..n} C(n+6, j+6)*C(j+6, 6)*(1+(-1)^j)/2.
G.f.: (1 - 7*x + 42*x^2 - 140*x^3 + 280*x^4 - 336*x^5 + 224*x^6 - 64*x^7)/ (1-2*x)^7.
E.g.f.: (x^6/6!)*exp(x)*cosh(x) (with 6 leading zeros).
a(n) = ceiling(binomial(n+6,6)*2^(n-1)). - Zerinvary Lajos, Nov 01 2006
From Amiram Eldar, Jan 07 2022: (Start)
Sum_{n>=0} 1/a(n) = 89/5 - 24*log(2).
Sum_{n>=0} (-1)^n/a(n) = 5832*log(3/2) - 11819/5. (End)

A082138 A transform of C(n,3).

Original entry on oeis.org

1, 4, 20, 80, 280, 896, 2688, 7680, 21120, 56320, 146432, 372736, 931840, 2293760, 5570560, 13369344, 31752192, 74711040, 174325760, 403701760, 928514048, 2122317824, 4823449600, 10905190400, 24536678400, 54962159616, 122607894528

Offset: 0

Paul Barry, Apr 06 2003

Fourth row of number array A082137. C(n,3) has e.g.f. (x^3/3!)exp(x). The transform averages the binomial and inverse binomial transforms.

			a(0) = (2^(-1) + 0^0/2)*C(3,0) = 2*(1/2) = 1 (using 0^0=1).

Vincenzo Librandi, Table of n, a(n) for n = 0..400
Index entries for linear recurrences with constant coefficients, signature (8,-24,32,-16).

Cf. A080929, A082137, A082139, A082140, A082141, A080951, A057711.

a:=[4,20,80,280];; for n in [5..30] do a[n]:=8*a[n-1]-24*a[n-2] +32*a[n-3]-16*a[n-4]; od; Concatenation([1], a);

[(Ceiling(Binomial(n+3, 3)*2^(n-1))) : n in [0..30]]; // Vincenzo Librandi, Sep 22 2011

[seq (ceil(binomial(n+3,3)*2^(n-1)),n=0..30)]; # Zerinvary Lajos, Nov 01 2006

Join[{1}, LinearRecurrence[{8,-24,32,-16}, {4,20,80,280}, 30]] (* G. C. Greubel, Jul 23 2019 *)

my(x='x+O('x^30)); Vec((1-4*x+12*x^2-16*x^3 + 8*x^4)/(1-2*x)^4) \\ G. C. Greubel, Jul 23 2019

((1-4*x+12*x^2-16*x^3+8*x^4)/(1-2*x)^4).series(x, 30).coefficients(x, sparse=False) # G. C. Greubel, Jul 23 2019

a(n) = (2^(n-1) + 0^n/2)*C(n+3, n).
a(n) = Sum_{j=0..n} C(n+3, j+3)*C(j+3, 3)*(1 + (-1)^j)/2.
G.f.: (1 - 4*x + 12*x^2 - 16*x^3 + 8*x^4)/(1-2*x)^4.
E.g.f.: (x^3/3!)*exp(x)*cosh(x) (preceded by 3 zeros).
a(n) = ceiling(binomial(n+3,3)*2^(n-1)). - Zerinvary Lajos, Nov 01 2006
From Amiram Eldar, Jan 07 2022: (Start)
Sum_{n>=0} 1/a(n) = 12*log(2) - 7.
Sum_{n>=0} (-1)^n/a(n) = 108*log(3/2) - 43. (End)

A054323 Fifth column of Lanczos triangle A053125 (decreasing powers).

Original entry on oeis.org

5, 140, 2016, 21120, 183040, 1397760, 9748480, 63504384, 392232960, 2321285120, 13264486400, 73610035200, 398475657216, 2111580405760, 10984378859520, 56221121904640, 283661115064320, 1413061420253184, 6959221409054720

Offset: 0

Wolfdieter Lang

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.

G. C. Greubel, Table of n, a(n) for n = 0..1000
Index entries for sequences related to Chebyshev polynomials.
Index entries for linear recurrences with constant coefficients, signature (20, -160, 640, -1280, 1024).

Cf. A053125, A054322.

Bisection of A080951.

List([0..20], n-> 4^n*Binomial(2*n+5, 4)); # G. C. Greubel, Jul 22 2019

[4^n*Binomial(2*n+5, 4): n in [0..20]]; // G. C. Greubel, Jul 22 2019

Table[4^n Binomial[2n+5,4],{n,0,20}] (* or *) LinearRecurrence[{20,-160, 640,-1280,1024},{5,140,2016,21120,183040},20] (* Harvey P. Dale, Mar 03 2018 *)

vector(20, n, n--; 4^n*binomial(2*n+5, 4)) \\ G. C. Greubel, Jul 22 2019

[4^n*binomial(2*n+5, 4) for n in (0..20)] # G. C. Greubel, Jul 22 2019

a(n) = 4^n*binomial(2*n+5, 4) = 4^n*A053126(n+4) = A053125(n+4, 4).
G.f.: (5 +40*x +16*x^2)/(1-4*x)^5.
E.g.f.: (15 +360*x +1464*x^2 +1664*x^3 +512*x^4)*exp(4*x)/3. - G. C. Greubel, Jul 22 2019
a(n) = 20*a(n-1)-160*a(n-2)+640*a(n-3)-1280*a(n-4)+1024*a(n-5). - Wesley Ivan Hurt, May 02 2021

cp's OEIS Frontend

A057711 a(0)=0, a(1)=1, a(n) = n*2^(n-2) for n >= 2.

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A082137 Square array of transforms of binomial coefficients, read by antidiagonals.

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A080929 Sequence associated with a(n) = 2*a(n-1) + k*(k+2)*a(n-2).

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A082139 A transform of binomial(n,5).

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A082140 A transform of binomial(n,6).

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A082138 A transform of C(n,3).

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A080929 Sequence associated with a(n) = 2a(n-1) + k(k+2)*a(n-2).