A253283 -id:A253283 - cp's OEIS frontend

A002002 a(n) = Sum_{k=0..n-1} binomial(n,k+1) * binomial(n+k,k).

0, 1, 5, 25, 129, 681, 3653, 19825, 108545, 598417, 3317445, 18474633, 103274625, 579168825, 3256957317, 18359266785, 103706427393, 586889743905, 3326741166725, 18885056428537, 107347191941249, 610916200215241

Offset: 0

N. J. A. Sloane, Simon Plouffe

From Benoit Cloitre, Jan 29 2002: (Start)
Array interpretation (first row and column are the natural numbers):
  1 2 3 ..j ... if b(i,j) = b(i-1,j) + b(i-1,j-1) + b(i,j-1) then a(n+1) = b(n,n)
  2 5 .........
  .............
  i........... b(i,j)
(End)
Number of ordered trees with 2n edges, having root of even degree, nonroot nodes of outdegree at most 2 and branches of odd length. - Emeric Deutsch, Aug 02 2002
Coefficient of x^n in ((1-x)/(1-2x))^n, n>0. - Michael Somos, Sep 24 2003
Number of peaks in all Schroeder paths (i.e., consisting of steps U=(1,1), D=(1,-1), H=(2,0) and never going below the x-axis) from (0,0) to (2n,0). Example: a(2)=5 because HH, HU*D, U*DH, UHD, U*DU*D, UU*DD contain 5 peaks (indicated by *). - Emeric Deutsch, Dec 06 2003
a(n) is the total number of HHs in all Schroeder (n+1)-paths. Example: a(2)=5 because UH*HD, H*H*H, UDH*H, H*HUD contain 5 HHs (indicated by *) and the other 18 Schroeder 3-paths contain no HHs. - David Callan, Jul 03 2006
a(n) is the total number of Hs in all Schroeder n-paths. Example: a(2)=5 as the Schroeder 2-paths are HH, DUH, DHU, HDU, DUDU and DDUU, and there are 5 H's. In general, a(n) is the total number of H..Hs (m+1 H's) in all Schroeder (n+m)-paths. - FUNG Cheok Yin, Jun 19 2021
a(n) is the number of points in Z^(n+1) that are L1 (Manhattan) distance <= n from the origin, or the number of points in Z^n that are L1 distance <= n+1 from the origin. These terms occur in the crystal ball sequences: a(n) here is the n-th term in the sequence for the (n+1)-dimensional cubic lattice as well as the (n+1)-st term in the sequence for the n-dimensional cubic lattice. See A008288 for a list of crystal ball sequences (rows or columns of A008288). - Shel Kaphan, Dec 25 2022 [Edited by Peter Munn, Jan 05 2023]

			G.f. = x + 5*x^2 + 25*x^3 + 129*x^4 + 681*x^5 + 3653*x^6 + 19825*x^7 + 108545*x^8 + ...

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).

G. C. Greubel, Table of n, a(n) for n = 0..1000 (terms 0 through 100 were computed by Vincenzo Librandi)
Paul Barry, On Integer-Sequence-Based Constructions of Generalized Pascal Triangles, Journal of Integer Sequences, Vol. 9 (2006), Article 06.2.4.
Paul Barry, Riordan arrays, generalized Narayana triangles, and series reversion, Linear Algebra and its Applications, 491 (2016) 343-385.
Paul Barry, On the inversion of Riordan arrays, arXiv:2101.06713 [math.CO], 2021.
F. D. Cunden, F. Mezzadri, N. Simm and P. Vivo, Correlators for the Wigner-Smith time-delay matrix of chaotic cavities, arXiv:1601.06690 [math-ph], 2016.
L. Ericksen, Lattice path combinatorics for multiple product identities, J. of Statistical Planning and Inference 140 (2010) 2113-2226, see p. 2219.
Ricardo Gómez Aíza, Trees with flowers: A catalog of integer partition and integer composition trees with their asymptotic analysis, arXiv:2402.16111 [math.CO], 2024. See p. 19.
Milan Janjic, Two Enumerative Functions
Milan Janjić, On Restricted Ternary Words and Insets, arXiv:1905.04465 [math.CO], 2019.
Milan Janjic and B. Petkovic, A Counting Function, arXiv:1301.4550 [math.CO], 2013.
G. Rutledge and R. D. Douglass, Integral functions associated with certain binomial coefficient sums, Amer. Math. Monthly, 43 (1936), 27-32.

Bisection of A002003, Cf. A047781, A001003.
a(n)=T(n, n+1), array T as in A050143.
a(n)=T(n, n+1), array T as in A064861.
Half the first differences of central Delannoy numbers (A001850).
a(n)=T(n, n+1), array T as in A008288.
Cf. A026002, A190666, A259554.

[&+[Binomial(n,k+1)*Binomial(n+k,k): k in [0..n]]: n in [0..21]];  // Bruno Berselli, May 19 2011

A064861 := proc(n,k) option remember; if n = 1 then 1; elif k = 0 then 0; else A064861(n,k-1)+(3/2-1/2*(-1)^(n+k))*A064861(n-1,k); fi; end; seq(A064861(i,i+1),i=1..40);

CoefficientList[Series[((1-x)/Sqrt[1-6x+x^2]-1)/2, {x,0,30}],x]  (* Harvey P. Dale, Mar 17 2011 *)
a[ n_] := n Hypergeometric2F1[ n + 1, -n + 1, 2, -1] (* Michael Somos, Aug 09 2011 *)
a[ n_] := With[{m = Abs@n}, Sign[n] Sum[ Binomial[ m, k] Binomial[ m + k - 1, m], {k, m}]]; (* Michael Somos, Aug 09 2011 *)

makelist(sum(binomial(n,k+1)*binomial(n+k,k), k, 0, n), n, 0, 21); /* Bruno Berselli, May 19 2011 */

{a(n) = my(m = abs(n)); sign( n) * sum( k=0, m-1, binomial( m, k+1) * binomial( m+k, k))}; /* Michael Somos, Aug 09 2011 */

/* L.g.f.: Sum_{n>=1} d^(n-1)/dx^(n-1) x^(2*n-1)*(1-x)^(-n)/n! */
{Dx(n, F)=local(D=F); for(i=1, n, D=deriv(D)); D}
{a(n)=local(A=1); A=(sum(m=1, n+1, Dx(m-1, x^(2*m-1)/(1-x)^m/m!)+x*O(x^n))); n*polcoeff(A, n)}
for(n=0, 30, print1(a(n), ", ")) \\ Paul D. Hanna, May 17 2015

a = lambda n: hypergeometric([1-n, -n], [1], 2) if n>0 else 0
[simplify(a(n)) for n in range(22)] # Peter Luschny, Nov 19 2014

G.f.: ((1-x)/sqrt(1-6*x+x^2)-1)/2. - Emeric Deutsch, Aug 02 2002
E.g.f.: exp(3*x)*(BesselI(0, 2*sqrt(2)*x)+sqrt(2)*BesselI(1, 2*sqrt(2)*x)). - Vladeta Jovovic, Mar 28 2004
a(n) = Sum_{k=0..n-1} binomial(n-1, k)*binomial(n+k, k+1). - Paul Barry, Sep 20 2004
a(n) = n * hypergeom([n + 1, -n + 1], [2], -1) = ((n+1)*LegendreP(n+1,3) - (5*n+3)*LegendreP(n,3))/(2*n) for n > 0. - Mark van Hoeij, Jul 12 2010
G.f.: x*d/dx log(1/(1-x*A006318(x))). - Vladimir Kruchinin, Apr 19 2011
a(n) = -a(-n) for all n in Z. - Michael Somos, Aug 09 2011
G.f.: -1 + 1 / ( 1 - x / (1 - 4*x / (1 - x^2 / (1 - 4*x / (1 - x^2 / (1 - 4*x / ...)))))). - Michael Somos, Jan 03 2013
a(n) = Sum_{k=0..n} A201701(n,k)^2 = Sum_{k=0..n} A124182(n,k)^2 for n > 0. - Philippe Deléham, Dec 05 2011
D-finite with recurrence: 2*(6*n^2-12*n+5)*a(n-1)-(n-2)*(2*n-1)*a(n-2)-n*(2*n-3)*a(n)=0. - Vaclav Kotesovec, Oct 04 2012
a(n) ~ (3+2*sqrt(2))^n/(2^(5/4)*sqrt(Pi*n)). - Vaclav Kotesovec, Oct 04 2012
D-finite (an alternative): n*a(n) = (6-n)*a(n-6) + (14*n-72)*a(n-5) + (264-63*n)*a(n-4) + 100*(n-3)*a(n-3) + (114-63*n)*a(n-2) + 2*(7*n-6)*a(n-1), n >= 7. - Fung Lam, Feb 05 2014
a(n) = (-1)^(n-1)*Sum_{k=0..n-1} (-2)^k*binomial(n-1,k)*binomial(n+k,k) and n^3*a(n) = Sum_{k=0..n-1} (4*k^3+4*k^2+4*k+1)*binomial(n-1,k)*binomial(n+k,k). For each of the two equalities, both sides satisfy the same recurrence -- this follows from the Zeilberger algorithm. - Zhi-Wei Sun, Aug 30 2014
a(n) = hypergeom([1-n, -n], [1], 2) for n >= 1. - Peter Luschny, Nov 19 2014
Logarithmic derivative of A001003 (little Schroeder numbers). - Paul D. Hanna, May 17 2015
L.g.f.: L(x) = Sum_{n>=1} d^(n-1)/dx^(n-1) x^(2*n-1) * (1-x)^(-n) / n! = Sum_{n>=1} a(n)*x^n/n where exp(L(x)) = g.f. of A001003. - Paul D. Hanna, May 17 2015
a(n+1) = (1/2^(n+1)) * Sum_{k >= 0} (1/2^k) * binomial(n + k, n)*binomial(n + k, n + 1). - Peter Bala, Mar 02 2017
2*a(n) = A110170(n), n > 0. - R. J. Mathar, Feb 10 2022
a(n) = (LegendreP(n,3) - LegendreP(n-1,3))/2. - Mark van Hoeij, Jul 14 2022
D-finite with recurrence n*a(n) +(-7*n+5)*a(n-1) +(7*n-16)*a(n-2) +(-n+3)*a(n-3)=0. - R. J. Mathar, Aug 01 2022
From Peter Bala, Nov 08 2022: (Start)
a(n) = (-1)^(n+1)*hypergeom( [n+1, -n+1], [1], 2) for n >= 1.
The Gauss congruences hold: a(n*p^r) == a(n^p^(r-1)) (mod p^r) for all primes p and all positive integers n and r. (End)
From Peter Bala, Apr 18 2024: (Start)
G.f.: Sum_{n >= 1} binomial(2*n-1, n)*x^n/(1 - x)^(2*n) = x + 5*x^2 + 25*x^3 + 129*x^4 + ....
Row sums of A253283. (End)

More terms from Clark Kimberling

A182626 a(n) = Hypergeometric([-n, n], [1], 2).

Original entry on oeis.org

1, -1, 5, -25, 129, -681, 3653, -19825, 108545, -598417, 3317445, -18474633, 103274625, -579168825, 3256957317, -18359266785, 103706427393, -586889743905, 3326741166725, -18885056428537, 107347191941249, -610916200215241

Offset: 0

Michael Somos, Feb 06 2011

			G.f. = 1 - x + 5*x^2 - 25*x^3 + 129*x^4 - 681*x^5 + 3653*x^6 - 19825*x^7 + ...

Vincenzo Librandi, Table of n, a(n) for n = 0..1000

Cf. A002002, A026003, A253283, A001003, A178301.

m:=30; R:=PowerSeriesRing(Rationals(), m); Coefficients(R!(1/2 + 1/2*(1 + x)/Sqrt(1 + 6*x + x^2))); // G. C. Greubel, Aug 14 2018

seq(simplify(hypergeom([-n, n],[1],2)), n=0..21); # Peter Luschny, Mar 23 2015

a[n_] := Hypergeometric2F1[ -n, n, 1, 2]; Array[a, 20, 0]

{a(n) = sum( k=0, abs(n), 2^k * prod( i=0, k-1, i^2 - n^2 ) / k!^2)}

a(-n) = a(n). a(n) = (-1)^n * A002002(n) if n>0. a(n) = (-1)^n * A026003(2*n - 1) if n>0.
G.f.: 1 / ( 1 + x / (1 + 4*x / (1 - x^2 / (1 + 4*x / (1 - x^2 / (1 + 4*x / ...)))))). - Michael Somos, Jan 03 2013
a(n) = (-1)^n*Sum_{k=0..n} A253283(n,k). - Peter Luschny, Mar 23 2015
From Peter Bala, Jun 17 2015: (Start)
a(n) = Sum_{k = 0..n} (-2)^k*binomial(n,k)*binomial(n+k-1,k) = (-1)^n*Sum_{k = 0..n-1} binomial(n,k+1)*binomial(n+k,k) = -Sum_{k = 0..n-1} (-2)^k*binomial(n-1,k)*binomial(n+k,k).
a(n) = -R(n-1,-2) for n >= 1, where R(n,x) denotes the n-th row polynomial of A178301.
a(n) = [x^n] ((x - 1)/(1 - 2*x))^n. Cf. A001003(n) =  (-1)^(n+1)/(n+1)*[x^n] ((x - 1)/(1 - 2*x))^(n+1).
O.g.f.: 1/2 + 1/2*(1 + x)/sqrt(1 + 6*x + x^2).
exp( Sum_{n >= 1} a(n)*(-x)^n/n ) = 1 + x + 3*x^2 + 11*x^3 + 45*x^4 + ... is the o.g.f. for A001003.
Recurrence: n*(3 - 2*n )*a(n) = 2*(6*n^2 - 12*n + 5)*a(n-1) + (2*n - 1)*(n - 2)*a(n-2) with a(0) = 1, a(1) = -1. (End)

A208881 Number of words either empty or beginning with the first letter of the ternary alphabet, where each letter of the alphabet occurs n times.

Original entry on oeis.org

1, 2, 30, 560, 11550, 252252, 5717712, 133024320, 3155170590, 75957810500, 1850332263780, 45508998487680, 1128243920840400, 28159366024288800, 706857555303576000, 17831659928458210560, 451781821468671694110, 11489952898943726476500, 293206575828601020085500

Offset: 0

Alois P. Heinz, Mar 02 2012

Also the number of (n*k-1)-step walks on k-dimensional cubic lattice from (1,0,...,0) to (n,n,...,n) with positive unit steps in all dimensions.

			a(0) = 1: the empty word.
a(1) = 2 = |{abc, acb}|.
a(2) = 30 = |{aabbcc, aabcbc, aabccb, aacbbc, aacbcb, aaccbb, ababcc, abacbc, abaccb, abbacc, abbcac, abbcca, abcabc, abcacb, abcbac, abcbca, abccab, abccba, acabbc, acabcb, acacbb, acbabc, acbacb, acbbac, acbbca, acbcab, acbcba, accabb, accbab, accbba}|.

Alois P. Heinz, Table of n, a(n) for n = 0..220
J. Draisma, E. Horobet, G. Ottaviani, B. Sturmfels and R. K. Thomas, The Euclidean distance degree of an algebraic variety, arXiv preprint arXiv: 1309.0049, 2013.

Column k=3 of A208879.

Cf. A060542, A253283.

a:= n-> `if`(n=0, 1, (3*n)!/(3*n!^3)):
seq(a(n), n=0..20);

a(n) = (3*n)!/(3 * n!^3) for n>0, a(0) = 1.
a(n) = 2 * A060542(n) for n>0.
a(n) = A253283(2*n,n) for n>=0. - Peter Luschny, Mar 22 2015
n^2*a(n) -3*(3*n-1)*(3*n-2)*a(n-1)=0. - R. J. Mathar, Nov 01 2015

A266732 a(n) = 10*binomial(n+4, 5).

Original entry on oeis.org

0, 10, 60, 210, 560, 1260, 2520, 4620, 7920, 12870, 20020, 30030, 43680, 61880, 85680, 116280, 155040, 203490, 263340, 336490, 425040, 531300, 657800, 807300, 982800, 1187550, 1425060, 1699110, 2013760, 2373360, 2782560, 3246320, 3769920, 4358970, 5019420

Offset: 0

Alan Shore and N. J. A. Sloane, Jan 06 2016

Total number of pips on a set of tetrominoes (4-celled linear dominoes) with up to n pips in each cell.

Colin Barker, Table of n, a(n) for n = 0..1000
Steve Butler and Pavel Karasik, A note on nested sums, J. Int. Seq. (2010) Vol. 13, Issue 4, Art. No. 10.4.4. See p=4 in the last equation on page 3.
Sela Fried, Counting r X s rectangles in nondecreasing and Smirnov words, arXiv:2406.18923 [math.CO], 2024. See p. 9.
Index entries for linear recurrences with constant coefficients, signature (6,-15,20,-15,6,-1).

Row 4 of array in A129533. Column k=3 in A253283.

[10*Binomial(n+4,5): n in [0..30]]; // G. C. Greubel, Nov 24 2017

Join[{0},10*Binomial[Range[0,40]+5,5]] (* or *) LinearRecurrence[{6,-15,20,-15,6,-1},{0,10,60,210,560,1260},40] (* Harvey P. Dale, Jun 10 2016 *)

a(n) = (n*(1+n)*(2+n)*(3+n)*(4+n))/12 \\ Colin Barker, Jan 08 2016

concat(0, Vec(10*x/(1-x)^6 + O(x^50))) \\ Colin Barker, Jan 08 2016

From Colin Barker, Jan 08 2016: (Start)
a(n) = n*(1+n)*(2+n)*(3+n)*(4+n)/12.
a(n) = 6*a(n-1) - 15*a(n-2) + 20*a(n-3) - 15*a(n-4) + 6*a(n-5) - a(n-6) for n > 5.
G.f.: 10*x / (1-x)^6.
(End)
a(n) = 10*A000389(n+4). - R. J. Mathar, Dec 18 2016
E.g.f.: x*(120 + 240*x + 120*x^2 + 20*x^3 + x^4)*exp(x)/12. - G. C. Greubel, Nov 24 2017

A299507 a(n) = (-1)^n*hypergeom([-n, n], [1], 4).

Original entry on oeis.org

1, 3, 33, 387, 4737, 59523, 761121, 9854211, 128772609, 1694927619, 22437369633, 298419470979, 3984500221569, 53376363001731, 717044895641121, 9656091923587587, 130310873022310401, 1761872309456567811, 23861153881099854369, 323634591584064809859

Offset: 0

Peter Luschny, Mar 16 2018

nonn

Cf. A299864, A245926, A084768, A245927, A253283.

Cf. A300945, A300946.

seq(simplify( (-1)^n*hypergeom([-n, n], [1], 4)), n = 0..20); # Peter Bala, Apr 18 2024

a[n_] := (-1)^n Hypergeometric2F1[-n, n, 1, 4]; Table[a[n], {n, 0, 19}]

From Vaclav Kotesovec, Jul 05 2018: (Start)
Recurrence: n*(2*n-3)*a(n) = 2*(14*n^2 - 28*n + 11)*a(n-1) - (n-2)*(2*n-1)*a(n-2).
a(n) ~ 2^(-3/2) * 3^(1/4) * (7 + 4*sqrt(3))^n / sqrt(Pi*n). (End)
From Peter Bala, Apr 18 2024: (Start)
a(n) = Sum_{k = 0..n} binomial(n, k)*binomial(n+k-1, k-1)*3^k = R(n, 3) for n >= 1, where R(n, x) denotes the n-th row polynomial of A253283.
a(n) = 3*n* hypergeom([1 - n, n + 1], [2], -3) for n >= 1.
a(n) = (1/2)*(LegendreP(n, 7) - LegendreP(n-1, 7)) for n >= 1.
a(n) = [x^n] ( (1 - x)/(1 - 4*x) )^n.
It follows that the Gauss congruences hold: a(n*p^r) == a(n*p^(r-1)) (mod p^r) for all primes p and positive integers n and r.
G.f.: (sqrt(x^2 - 14*x + 1) - x + 1)/(2*sqrt(x^2 - 14*x + 1)) = 1 + 3*x + 33*x^2 + 387*x^3 + .... (End)

A303700 Triangle read by rows in which row n gives coefficients of polynomial f_n(x)/(n+1) of degree less than n that satisfies Integral_{x=0..1} g(t - x) * f_n(x) dx = g(t) for any polynomial g(x) of degree less than n.

Original entry on oeis.org

1, 2, -3, 3, -12, 10, 4, -30, 60, -35, 5, -60, 210, -280, 126, 6, -105, 560, -1260, 1260, -462, 7, -168, 1260, -4200, 6930, -5544, 1716, 8, -252, 2520, -11550, 27720, -36036, 24024, -6435, 9, -360, 4620, -27720, 90090, -168168, 180180, -102960, 24310

Offset: 0

Seiichi Manyama, Apr 28 2018

			Triangle begins:
n | 0     1     2      3     4      5     6
--*-----------------------------------------
0 | 1;
1 | 2,   -3;
2 | 3,  -12,   10;
3 | 4,  -30,   60,   -35;
4 | 5,  -60,  210,  -280,  126;
5 | 6, -105,  560, -1260, 1260,  -462;
6 | 7, -168, 1260, -4200, 6930, -5544, 1716;

Seiichi Manyama, Rows n = 0..139, flattened

Cf. A253283, A303699.

f_n(x)/(n+1) = 1/(n!*x) * d^n/dx^n x^{n+1}*(1-x)^n.

T(n,k) = (-1)^(k)*(n+k+1)!*(k+1)/((k+1)!^2*(n-k)!). - Jacob Fauman, Sep 20 2022

A253284 Triangle read by rows, T(n,k) = (k+1)(n+1)!(n+k)!/((k+1)!^2*(n-k)!) with n >= 0 and 0 <= k <= n.

Original entry on oeis.org

1, 2, 2, 6, 18, 12, 24, 144, 240, 120, 120, 1200, 3600, 4200, 1680, 720, 10800, 50400, 100800, 90720, 30240, 5040, 105840, 705600, 2116800, 3175200, 2328480, 665280, 40320, 1128960, 10160640, 42336000, 93139200, 111767040, 69189120, 17297280

Offset: 0

Peter Luschny, Mar 23 2015

G_n(x) = - Sum_{k=0..n} T(n,k)/(x-1)^(n+k+1) are generating functions, for n=0 of A000012, for n=1 of A002378, for n=2 of A083374 (with offset 0) and for n=3 for A253285. In general G_n(x) is the generating function of the sequence k -> ((n+k)!/k!)*C(n+k-1,k-1). These sequences are associated with the rows of the square array of unsigned Lah numbers (compare A253283 for the columns).

			Triangle begins:
1;
2, 2;
6, 18, 12;
24, 144, 240, 120;
120, 1200, 3600, 4200, 1680;
720, 10800, 50400, 100800, 90720, 30240;
5040, 105840, 705600, 2116800, 3175200, 2328480, 665280.

Cf. A000142, A001804, A001813, A002378, A006318, A083374, A088617, A253283, A253285.

/* As triangle: */ [[(k + 1)*Factorial(n + 1)*Factorial(n + k)/(Factorial(k + 1)^2*Factorial(n - k)): k in [0..n]]: n in [0..10]]; // Bruno Berselli, Mar 23 2015

T := (n,k) -> ((k+1)*(n+1)!*(n+k)!)/((k+1)!^2*(n-k)!);
for n from 0 to 6 do seq(T(n,k), k=0..n) od;

f[n_] := Rest@ Flatten@ Reap@ Block[{i, k, t}, For[i = 0, i <= n, i++, For[k = 0, k <= i, k++, Sow[(i + 1)!*Binomial[i + k, i]*Binomial[i, k]/(k + 1)]]]]; f@ 7 (* Michael De Vlieger, Mar 23 2015 *)

tabl(nn) = {for (n=0, nn, for (k=0, n, print1((n+1)!*binomial(n+k,n)*binomial(n,k)/(k+1), ", ");); print(););} \\ Michel Marcus, Mar 23 2015

T(n,k) = (n+1)!*binomial(n+k,n)*binomial(n,k)/(k+1).
T(n,k) = (n+1)!*A088617(n,k).
T(n,0) = n! = A000142(n).
T(n,1) = A001804(n+1) for n>0.
T(n,n) = (2*n)!/n! = A001813(n).
Sum_{k=0..n} T(n,k) = (n+1)!*hypergeom([-n, n+1], [2], -1) = (n+1)!*A006318(n).

A345013 Triangle read by rows, related to clusters of type D.

Original entry on oeis.org

1, 4, 3, 15, 20, 6, 56, 105, 60, 10, 210, 504, 420, 140, 15, 792, 2310, 2520, 1260, 280, 21, 3003, 10296, 13860, 9240, 3150, 504, 28, 11440, 45045, 72072, 60060, 27720, 6930, 840, 36

Offset: 1

F. Chapoton, Sep 30 2021

Let C_{n+1} be the cyclic quiver with n+1 vertices. Empirically, the n-th row is related to the green-mutation partial order on clusters for this quiver, restricted to clusters that do not meet the initial seed.
Apparently, value of the associated polynomials at -2 is A089849, up to sign.
By evaluating the associated polynomials at x-1, one apparently gets A062196.
The rows seem to give (up to sign) the coefficients in the expansion of the integer-valued polynomial (x+1)^2*(x+2)^2*(x+3)^2*...*(x+n)^2*(x+n+1)*(x+n+2) / (n! * (n+2)!) in the basis made of the binomial(x+i,i). - F. Chapoton, Oct 31 2022
Chapoton's observation above is correct: the precise expansion is (x+1)^2*(x+2)^2*(x+3)^2*...*(x+n)^2*(x+n+1)*(x+n+2) / (n! * (n+2)!) = Sum_{k = 0..n} (-1)^k*T(n+1,k)*binomial(x+2*n+2-k, 2*n+2-k), as can be verified using the WZ algorithm. For example, n = 2 gives (x+1)^2*(x+2)^2*(x+3)*(x+4)/(2!*4!) = 15*binomial(x+6,6) - 20*binomial(x+5,5) + 6*binomial(x+4,4). - Peter Bala, Jun 24 2023

			Triangle begins:
[1] 1
[2] 4,    3
[3] 15,   20,    6
[4] 56,   105,   60,    10
[5] 210,  504,   420,   140,  15
[6] 792,  2310,  2520,  1260, 280,  21
[7] 3003, 10296, 13860, 9240, 3150, 504, 28
...

Cf. A001791 (T(n,1)), A000217 (T(n,n)), A026002 (row sums), A000012 (alternating row sum), A051924 (number of clusters of type D_n).

Cf. A089849, A062196, A063007, A253283.

row(n) = vector(n, k, k--; (n-k)*binomial(n,k)*binomial(2*n-k, n-1)/n); \\ Michel Marcus, Sep 30 2021

def T_row(n):
    return [(n-k)*binomial(n,k)*binomial(2*n-k,n-1)//n for k in range(n)]
for n in range(1, 8): print(T_row(n))

T(n, k) = (n-k)*binomial(n,k)*binomial(2*n-k, n-1)/n, for n >= 1 and 0 <= k < n.
From Peter Bala, Jun 24 2023: (Start)
As conjectured above by Chapoton we have
Sum_{k = 0..n-1} T(n,k)*(x - 1)^k = Sum_{k = 0..n-1} A062196(n-1,k)*x^k and
Sum_{k = 0..n-1} T(n,k)*(-2)^k = (-1)^floor(n/2)*A089849(n) for n >= 1 (both easily verified using the WZ algorithm). (End)

cp's OEIS Frontend

A002002 a(n) = Sum_{k=0..n-1} binomial(n,k+1) * binomial(n+k,k).

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A182626 a(n) = Hypergeometric([-n, n], [1], 2).

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A208881 Number of words either empty or beginning with the first letter of the ternary alphabet, where each letter of the alphabet occurs n times.

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A266732 a(n) = 10*binomial(n+4, 5).

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A299507 a(n) = (-1)^n*hypergeom([-n, n], [1], 4).

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A303700 Triangle read by rows in which row n gives coefficients of polynomial f_n(x)/(n+1) of degree less than n that satisfies Integral_{x=0..1} g(t - x) * f_n(x) dx = g(t) for any polynomial g(x) of degree less than n.

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A253284 Triangle read by rows, T(n,k) = (k+1)*(n+1)!*(n+k)!/((k+1)!^2*(n-k)!) with n >= 0 and 0 <= k <= n.

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A253284 Triangle read by rows, T(n,k) = (k+1)(n+1)!(n+k)!/((k+1)!^2*(n-k)!) with n >= 0 and 0 <= k <= n.