A002457 -id:A002457 - cp's OEIS frontend

A001810 a(n) = n!n(n-1)*(n-2)/36.

0, 0, 0, 1, 16, 200, 2400, 29400, 376320, 5080320, 72576000, 1097712000, 17563392000, 296821324800, 5288816332800, 99165306240000, 1952793722880000, 40311241850880000, 870722823979008000, 19645683716026368000, 462251381553561600000, 11325158848062259200000

Offset: 0

N. J. A. Sloane

nonn

a(n) is the total number of 3-2-1 patterns in all permutations on [n]. This is because there are n! permutations, binomial(n,3) triples in each one and the probability that a given triple of entries in a random permutation form a 3-2-1 pattern (or any other specified pattern of length 3) is 1/6. - David Callan, Oct 26 2006

Old name was "Coefficients of Laguerre polynomials".

			G.f. = x^3 + 16*x^4 + 200*x^5 + 2400*x^6 + 29400*x^7 + 376320*x^8 + ...

M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards Applied Math. Series 55, 1964 (and various reprintings), p. 799.
Cornelius Lanczos, Applied Analysis. Prentice-Hall, Englewood Cliffs, NJ, 1956, p. 519.
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).

T. D. Noe, Table of n, a(n) for n = 0..100
M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards, Applied Math. Series 55, Tenth Printing, 1972 [alternative scanned copy].
Cornelius Lanczos, Applied Analysis. (Annotated scans of selected pages)
Index entries for sequences related to Laguerre polynomials.

Cf. A053495, A263771.

Cf. A001113, A001620, A091725, A099285.

[Factorial(n)*n*(n-1)*(n-2)/36: n in [0..20]]; // G. C. Greubel, May 16 2018

[seq(n!*n*(n-1)*(n-2)/36,n=0..30)];
with(combstruct):ZL:=[st, {st=Prod(left, right), left=Set(U, card=r+1), right=Set(U, card=1)}, labeled]: subs(r=2, stack): seq(count(subs(r=2, ZL), size=m), m=0..20) ; # Zerinvary Lajos, Feb 07 2008

Table[n! n*(n-1)*(n-2)/36, {n, 0, 20}] (* T. D. Noe, Aug 10 2012 *)

for(n=0,20, print1(n!*n*(n-1)*(n-2)/36, ", ")) \\ G. C. Greubel, May 16 2018

[factorial(m) * binomial(m, 3) / 6 for m in range(22)]  # Zerinvary Lajos, Jul 05 2008

a(n) = -A021009(n, 3), n >= 0. a(n) = ((n!/3!)^2)/(n-3)!, n >= 3.
E.g.f.: x^3/(3!*(1-x)^4).
If we define f(n,i,x) = Sum_{k=i..n} Sum_{j=i..k} binomial(k,j) * Stirling1(n,k) * Stirling2(j,i) * x^(k-j) then a(n) = (-1)^(n-1) * f(n,3,-4), (n >= 3). - Milan Janjic, Mar 01 2009
a(n) = Sum_{k>0} k * A263771(n,k). - Alois P. Heinz, Oct 27 2015
From Amiram Eldar, May 02 2022: (Start)
Sum_{n>=3} 1/a(n) = 9*(2*e + gamma - Ei(1) - 4), where e = A001113, gamma = A001620, and Ei(1) = A091725.
Sum_{n>=3} (-1)^(n+1)/a(n) = 63*(gamma - Ei(-1)) - 36*(1/e + 1), where Ei(-1) = -A099285. (End)

Edited by N. J. A. Sloane, Apr 12 2014

A002011 a(n) = 4*(2n+1)!/n!^2.

Original entry on oeis.org

4, 24, 120, 560, 2520, 11088, 48048, 205920, 875160, 3695120, 15519504, 64899744, 270415600, 1123264800, 4653525600, 19234572480, 79342611480, 326704870800, 1343120024400, 5513861152800, 22606830726480, 92580354403680, 378737813469600

Offset: 0

N. J. A. Sloane, Simon Plouffe

nonn

R. C. Mullin, E. Nemeth and P. J. Schellenberg, The enumeration of almost cubic maps, pp. 281-295 in Proceedings of the Louisiana Conference on Combinatorics, Graph Theory and Computer Science. Vol. 1, edited R. C. Mullin et al., 1970.
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).

T. D. Noe, Table of n, a(n) for n = 0..200
Milan Janjić, Pascal Matrices and Restricted Words, J. Int. Seq., Vol. 21 (2018), Article 18.5.2.

a(n)=4 A002457(n).
a(n) = 2 * A005430(n+1) = 4 * A002457(n).
Cf. A001803.

seq(2*n*binomial(2*n,n), n=1..23); # Zerinvary Lajos, Dec 14 2007

Table[4*(2*n + 1)!/n!^2, {n, 0, 20}] (* T. D. Noe, Aug 30 2012 *)

PARI
```
a(n)=if(n<0,0,4*(2*n+1)!/n!^2)
```

G.f.: 4*(1-4x)^(-3/2).

a(n) = 1/J(n) where J(n) = Integral_{t=0..Pi/4} (cos(t)^2 - 1/2)^(2n+1). - Benoit Cloitre, Oct 17 2006

Simpler description from Travis Kowalski (tkowalski(AT)coloradocollege.edu), Mar 20 2003

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)

A033504 a(n)/4^n is the expected number of tosses of a coin required to obtain n+1 heads or n+1 tails.

Original entry on oeis.org

1, 10, 66, 372, 1930, 9516, 45332, 210664, 960858, 4319100, 19188796, 84438360, 368603716, 1598231992, 6889682280, 29551095248, 126193235194, 536799072924, 2275560109868, 9616650989560, 40527780684972, 170368957887656, 714556104675736, 2990728476330672

Offset: 0

Michael Ulm (ulm(AT)mathematik.uni-ulm.de)

The number of rooted two-vertex n-edge maps in the plane (planar with a distinguished outside face). - Valery A. Liskovets, Mar 17 2005

			From _Jeremy Tan_, Mar 13 2018: (Start)
For n=1 the sequences of flips ending at two heads or two tails are:
HH, TT (probability 1/4 each)
HTH, HTT, THH, THT (1/8 each)
The expected number of flips is 2*2*1/4 + 3*4*1/8 = 10/4 = a(1)/4^1. (End)

M. Klamkin, ed., Problems in Applied Mathematics: Selections from SIAM Review, SIAM, 1990; see pp. 127-129.
V. A. Liskovets and T. R. Walsh, Enumeration of unrooted maps on the plane, Rapport technique, UQAM, No. 2005-01, Montreal, Canada, 2005.

Vincenzo Librandi, Table of n, a(n) for n = 0..172
V. A. Liskovets and T. R. Walsh, Counting unrooted maps on the plane, Advances in Applied Math., 36, No.4 (2006), 364-387.

Cf. A002457, A100511, A103943.

Cf. A000346, A130783.

[(n+1)*(2^(2*n+1)-Binomial(2*n+1,n+1)): n in [0..25]]; // Vincenzo Librandi, Jun 09 2011

a[n_]:=(n+1)*(2^(2*n+1)-Binomial[2*n+1,n+1])
a /@ Range[0,50] (* Julien Kluge, Jul 21 2016 *)

With a different offset: Sum_{j=0..n} Sum_{k=0..n} binomial(n, j)*binomial(n, k)*min(j, k) = n*2^(n-1) + (n/2)*binomial(2*n, n). [see Klamkin]
a(n-1) = 4^(n-1)*b(n, n), where b(n, m) = b(n-1, m)/2 + b(n, m-1)/2 + 1; b(n, 0)=b(0, n)=0.
a(n) = Sum_{k=0..n, l=0..n} 2^(2n - k - l) binomial(k+l, k).
a(n) = (2n+1)*Sum_{0<=i,j<=n} binomial(2n, i+j)/(i+j+1). - Benoit Cloitre, Mar 05 2005
a(n) = (n+1)*(2^(2*n+1) - binomial(2*n+1,n+1)). - Vladeta Jovovic, Aug 23 2007
n*a(n) + 6*(-2*n+1)*a(n-1) + 48*(n-1)*a(n-2) + 32*(-2*n+3)*a(n-3) = 0. - R. J. Mathar, Dec 22 2013
a(n) ~ 2^(2*n+1)*n. - Ilya Gutkovskiy, Jul 21 2016

Name corrected by Jeremy Tan, Mar 13 2018

A045543 6-fold convolution of A000302 (powers of 4); expansion of 1/(1-4*x)^6.

Original entry on oeis.org

1, 24, 336, 3584, 32256, 258048, 1892352, 12976128, 84344832, 524812288, 3148873728, 18320719872, 103817412608, 574988746752, 3121367482368, 16647293239296, 87398289506304, 452414675091456, 2312341672689664, 11683410556747776, 58417052783738880, 289303499500421120

Offset: 0

Wolfdieter Lang

Also convolution of A020922 with A000984 (central binomial coefficients); also convolution of A040075 with A000302 (powers of 4).
With a different offset, number of n-permutations of 5 objects: u,v,z,x, y with repetition allowed, containing exactly five (5) u's. Example: a(1)=24 because we have uuuuuv uuuuvu uuuvuu uuvuuu uvuuuu vuuuuu uuuuuz uuuuzu uuuzuu uuzuuu uzuuuu zuuuuu uuuuux uuuuxu uuuxuu uuxuuu uxuuuu xuuuuu uuuuuy uuuuyu uuuyuu uuyuuu uyuuuu yuuuuu. - Zerinvary Lajos, Jun 16 2008
Also convolution of A002457 with A020920, also convolution of A002697 with A038846, also convolution of A002802 with A020918, also convolution of A038845 with A038845. - Rui Duarte, Oct 08 2011

Vincenzo Librandi, Table of n, a(n) for n = 0..400
Index entries for linear recurrences with constant coefficients, signature (24,-240,1280,-3840,6144,-4096).

Cf. A038231.

List([0..30], n-> 4^n*Binomial(n+5,5)); # G. C. Greubel, Jul 20 2019

[4^n*Binomial(n+5, 5): n in [0..30]]; // Vincenzo Librandi, Oct 15 2011

seq(seq(binomial(i+5, j)*4^i, j =i), i=0..30); # Zerinvary Lajos, Dec 03 2007
seq(binomial(n+5,5)*4^n,n=0..30); # Zerinvary Lajos, Jun 16 2008

CoefficientList[Series[1/(1-4x)^6,{x,0,30}],x] (* or *) LinearRecurrence[ {24,-240,1280,-3840,6144,-4096}, {1,24,336,3584,32256, 258048}, 30] (* Harvey P. Dale, Mar 24 2018 *)

Vec(1/(1-4*x)^6 + O(x^30)) \\ Michel Marcus, Aug 21 2015

[lucas_number2(n, 4, 0)*binomial(n,5)/2^10 for n in range(5, 35)] # Zerinvary Lajos, Mar 11 2009

a(n) = binomial(n+5, 5)*4^n.
G.f.: 1/(1-4*x)^6.
a(n) = Sum_{ i_1+i_2+i_3+i_4+i_5+i_6+i_7+i_8+i_9+i_10+i_11+i_12 = n} f(i_1)* f(i_2)*f(i_3)*f(i_4)*f(i_5)*f(i_6)*f(i_7)*f(i_8)*f(i_9)*f(i_10) *f(i_11)*f(i_12), with f(k)=A000984(k). - Rui Duarte, Oct 08 2011
E.g.f.: (15 + 120*x + 240*x^2 + 160*x^3 + 32*x^4)*exp(4*x)/3. - G. C. Greubel, Jul 20 2019
From Amiram Eldar, Mar 25 2022: (Start)
Sum_{n>=0} 1/a(n) = 1620*log(4/3) - 465.
Sum_{n>=0} (-1)^n/a(n) = 12500*log(5/4) - 8365/3. (End)

A061928 Array T(n,m) = 1/beta(n+1,m+1) read by antidiagonals.

Original entry on oeis.org

6, 12, 12, 20, 30, 20, 30, 60, 60, 30, 42, 105, 140, 105, 42, 56, 168, 280, 280, 168, 56, 72, 252, 504, 630, 504, 252, 72, 90, 360, 840, 1260, 1260, 840, 360, 90, 110, 495, 1320, 2310, 2772, 2310, 1320, 495, 110, 132, 660, 1980, 3960, 5544, 5544, 3960

Offset: 1

Frank Ellermann, May 22 2001

beta(n+1,m+1) = Integral_{x=0..1} x^n * (1-x)^m dx for real n, m.

			Antidiagonals:
   6,
  12, 12,
  20, 30, 20,
  30, 60, 60, 30,
  ...
Array:
   6  12  20   30   42
  12  30  60  105  168
  20  60 140  280  504
  30 105 280  630 1260
  42 168 504 1260 2772

G. Boole, A Treatise On The Calculus of Finite Differences, Dover, 1960, p. 26.

Rows: 1/b(n, 2): A002378, 1/b(n, 3): A027480, 1/b(n, 4): A033488. Diagonals: 1/b(n, n): A002457, 1/b(n, n+1) A005430, 1/b(n, n+2): A000917.

T(i, j)=A003506(i+1, j+1).

t[n_, m_] := 1/Beta[n+1, m+1]; Take[ Flatten[ Table[ t[n+1-m, m], {n, 1, 10}, {m, 1, n}]], 52] (* Jean-François Alcover, Oct 11 2011 *)

A(i,j)=if(i<1||j<1,0,1/subst(intformal(x^i*(1-x)^j),x,1)) /* Michael Somos, Feb 05 2004 */

A(i,j)=if(i<1||j<1,0,1/sum(k=0,i,(-1)^k*binomial(i,k)/(j+1+k))) /* Michael Somos, Feb 05 2004 */

from sympy import factorial as f
def T(n, m): return f(n + m + 1)/(f(n)*f(m))
for n in range(1, 11): print([T(m, n - m + 1) for m in range(1, n + 1)]) # Indranil Ghosh, Apr 29 2017

beta(n+1, m+1) = gamma(n+1)*gamma(m+1)/gamma(n+m+2) = n!*m!/(n+m+1)!.

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

A092437 Triangle read by rows, arising from enumeration of domino tilings of Aztec Pillow-like regions.

Original entry on oeis.org

1, 1, 1, 2, 1, 1, 5, 6, 6, 1, 1, 5, 13, 26, 30, 20, 1, 1, 5, 13, 41, 90, 140, 140, 70, 1, 1, 5, 13, 41, 121, 302, 560, 742, 630, 252

Offset: 0

Christopher Hanusa (chanusa(AT)math.washington.edu), Mar 24 2004

The rows are of lengths 1, 3, 5, 7, ...
Call the first row row 0 and entries starting from 0. Then entries i=0 through k in row k are A046717(i).
In row k, entry k+1 is sequence A092438 and entry k+2 is sequence A092439.
In row k, entry 2k-1 is A002457(k-1) and entry 2k is A000984(k).

			Triangle begins:
  1;
  1, 1, 2;
  1, 1, 5, 6, 6;
  1, 1, 5, 13, 26, 30, 20;
  ...

James Propp, Enumeration of matchings: problems and progress, pp. 255-291 in L. J. Billera et al., eds, New Perspectives in Algebraic Combinatorics, Cambridge, 1999 (see Problem 13).

James Propp, Publications and Preprints.
James Propp, Enumeration of matchings: problems and progress, in L. J. Billera et al. (eds.), New Perspectives in Algebraic Combinatorics.

Cf. A092438-A092443.

A155865 Triangle T(n,k) = (n-1)*binomial(n-2, k-1) for 1 <= k <= n-1, n >= 2, and T(n,0) = T(n,n) = 1 for n >= 0, read by rows.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 2, 2, 1, 1, 3, 6, 3, 1, 1, 4, 12, 12, 4, 1, 1, 5, 20, 30, 20, 5, 1, 1, 6, 30, 60, 60, 30, 6, 1, 1, 7, 42, 105, 140, 105, 42, 7, 1, 1, 8, 56, 168, 280, 280, 168, 56, 8, 1, 1, 9, 72, 252, 504, 630, 504, 252, 72, 9, 1, 1, 10, 90, 360, 840, 1260, 1260, 840, 360, 90, 10, 1

Offset: 0

Roger L. Bagula, Jan 29 2009

			Triangle begins:
  1;
  1, 1;
  1, 1,  1;
  1, 2,  2,   1;
  1, 3,  6,   3,   1;
  1, 4, 12,  12,   4,   1;
  1, 5, 20,  30,  20,   5,   1;
  1, 6, 30,  60,  60,  30,   6,   1;
  1, 7, 42, 105, 140, 105,  42,   7,  1;
  1, 8, 56, 168, 280, 280, 168,  56,  8, 1;
  1, 9, 72, 252, 504, 630, 504, 252, 72, 9, 1;
  ...
ConvOffs transform of (1, 1, 2, 3) = integers of row 4: (1, 3, 6, 3, 1). _Gary W. Adamson_, Jul 09 2012

G. C. Greubel, Rows n = 0..50 of the triangle, flattened

Cf. A002457 (T(2*n, n)), A155863, A155864.

A155865:= func< n,k | k eq 0 or k eq n select 1 else (n-1)*Binomial(n-2, k-1) >;
[A155865(n,k): k in [0..n], n in [0..12]]; // G. C. Greubel, Jun 04 2021

p[x_, n_] = If[n==0, 1, 1 + x^n + x*D[(x+1)^(n-1), {x, 1}]];
Flatten[Table[CoefficientList[ExpandAll[p[x, n]], x], {n, 0, 10}]]
(* or *)
q = 1;
c[n_, q_]= If[n<2, 1, Product[(i-1)^q, {i, 2, n}]];
T[n_, m_, q_]= c[n, q]/(c[m, q]*c[n-m, q]);
Flatten[Table[T[n, m, q], {n,0,12}, {m, 0, n}]] (* Roger L. Bagula, Mar 09 2010 *)

T(n, k) := if k = 0 or k = n then 1 else (n-1)*binomial(n-2, k-1)$ create_list(T(n, k), n, 0, 12, k, 0, n); /* Franck Maminirina Ramaharo, Dec 05 2018 */

def A155865(n,k): return 1 if (k==0 or k==n) else (n-1)*binomial(n-2, k-1)
flatten([[A155865(n,k) for k in (0..n)] for n in (0..12)]) # G. C. Greubel, Jun 04 2021

T(n, k) = coefficients of (p(n, x)), where p(n, x) = 1 + x^n + x*((d/dx) (x+1)^n) and T(0, 0) = 1.
Define c(n) = Product_{i=2..n} (i - 1), with c(0) = c(1) = 1. Then T(n,m) = c(n)/(c(m)*c(n-m)). - Roger L. Bagula, Mar 09 2010
The triangle is the ConvOffsStoT transform of the natural numbers prefaced with a 1. A row with n integers is the ConvOffs transform of a finite series of the first (n-1) terms in (1, 1, 2, 3, 4, ...).  See A214281 for definitions of the transform. - Gary W. Adamson, Jul 09 2012
Sum_{k=0..n} T(n, k) = 2 + A001787(n-1) - (3/4)*[n==0]. - R. J. Mathar, Jul 17 2012
From Franck Maminirina Ramaharo, Dec 05 2018: (Start)
T(n, k) = (n-1)*binomial(n-2, k-1) with T(n, 0) = T(n, n) = 1.
n-th row polynomial is (1/2)*(1 + (-1)^(2^n) + 2*x^n + (1 + (-1)^(2^n))*(n - 1)*x*(x + 1)^(n - 2)).
G.f.: 1/(1 - y) + 1/(1 - x*y) + x*y^2/(1 - (1 + x)*y)^2 - 1.
E.g.f.: exp(y) + exp(x*y) + x*(1 - (1 - (1 + x)*y)*exp((1 + x)*y))/(1 + x)^2  - 1. (End)
T(2*n, n) = A002457(n). - Alois P. Heinz, Dec 05 2018

Edited and name clarified by Franck Maminirina Ramaharo, Dec 04 2018

A258431 Sum over all peaks of Dyck paths of semilength n of the arithmetic mean of the x and y coordinates.

Original entry on oeis.org

0, 1, 5, 23, 102, 443, 1898, 8054, 33932, 142163, 592962, 2464226, 10209620, 42190558, 173962532, 715908428, 2941192472, 12065310083, 49428043442, 202249741418, 826671597572, 3375609654698, 13771567556012, 56138319705908, 228669994187432, 930803778591278

Offset: 0

Alois P. Heinz, May 29 2015

A Dyck path of semilength n is a (x,y)-lattice path from (0,0) to (2n,0) that does not go below the x-axis and consists of steps U = (1,1) and D = (1,-1). A peak of a Dyck path is any lattice point visited between two consecutive steps UD.

Alois P. Heinz, Table of n, a(n) for n = 0..1000
Paul Barry, A Note on Riordan Arrays with Catalan Halves, arXiv:1912.01124 [math.CO], 2019.
Wikipedia, Average, Arithmetic mean
Wikipedia, Lattice path

Cf. A000302, A000346, A000531, A002457, A002697, A002802, A029887.

A258431:= func< n | n eq 0 select 0 else (4^(n-1) + Factorial(2*n-1)/Factorial(n-1)^2)/2 >;
[A258431(n): n in [0..40]]; // G. C. Greubel, Mar 18 2023

a:= proc(n) option remember; `if`(n<3, [0, 1, 5][n+1],
       ((8*n-10)*a(n-1)-(16*n-24)*a(n-2))/(n-1))
    end:
seq(a(n), n=0..30);

a[0]=0; a[1]=1; a[2]=5;
a[n_]:= a[n]= (2*(4*n-5)*a[n-1] - 8*(2*n-3)*a[n-2])/(n-1);
Table[a[n], {n,0,30}] (* Jean-François Alcover, May 31 2018, from Maple *)

def A258431(n): return 0 if (n==0) else (4^(n-1) + factorial(2*n-1)/factorial(n-1)^2)/2
[A258431(n) for n in range(41)] # G. C. Greubel, Mar 18 2023

G.f.: x*(1 + sqrt(1-4*x))/(2*sqrt(1-4*x)^3).
a(n) = (2*(4*n-5)*a(n-1) - 8*(2*n-3)*a(n-2))/(n-1) for n>2, a(0)=0, a(1)=1, a(2)=5.
a(n) = (4^(n-1) + (2*n-1)!/(n-1)!^2)/2 for n>0, a(0) = 0.
a(n) = (A000302(n-1) + A002457(n-1))/2 for n>0, a(0) = 0.
a(n) = (1/2)*binomial(2*n,n)*( 1 + 2*(n-1)/(n+1) + 3*(n-1)*(n-2)/((n+1)*(n+2)) + 4*(n-1)*(n-2)*(n-3)/((n+1)*(n+2)*(n+3)) + 5*(n-1)*(n-2)*(n-3)*(n-4)/((n+1)*(n+2)*(n+3)*(n+4)) + ...) for n >= 1. - Peter Bala, Feb 17 2022

cp's OEIS Frontend

A001810 a(n) = n!*n*(n-1)*(n-2)/36.

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A002011 a(n) = 4*(2n+1)!/n!^2.

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

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A033504 a(n)/4^n is the expected number of tosses of a coin required to obtain n+1 heads or n+1 tails.

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A045543 6-fold convolution of A000302 (powers of 4); expansion of 1/(1-4*x)^6.

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A061928 Array T(n,m) = 1/beta(n+1,m+1) read by antidiagonals.

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A001810 a(n) = n!n(n-1)*(n-2)/36.

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