A214292 -id:A214292 - cp's OEIS frontend

A000344 a(n) = 5*binomial(2n, n-2)/(n+3).

1, 5, 20, 75, 275, 1001, 3640, 13260, 48450, 177650, 653752, 2414425, 8947575, 33266625, 124062000, 463991880, 1739969550, 6541168950, 24647883000, 93078189750, 352207870014, 1335293573130, 5071418015120, 19293438101000, 73514652074500, 280531912316292

Offset: 2

N. J. A. Sloane

a(n-3) is the number of n-th generation vertices in the tree of sequences with unit increase labeled by 4 (cf. Zoran Sunic reference). - Benoit Cloitre, Oct 07 2003
Number of lattice paths from (0,0) to (n,n) with steps E=(1,0) and N=(0,1) which touch but do not cross the line x-y=2. Example: For n=3 there are the 5 paths EENENN, EENNEN, EENNNE, ENEENN, NEEENN. - Herbert Kociemba, May 24 2004
Number of standard tableaux of shape (n+2,n-2). - Emeric Deutsch, May 30 2004

			G.f. = x^2 + 5*x^3 + 20*x^4 + 75*x^5 + 275*x^6 + 1001*x^7 + 3640*x^8 + ...

C. Krishnamachary and M. Bheemasena Rao, Determinants whose elements are Eulerian, prepared Bernoullian and other numbers, J. Indian Math. Soc., Vol. 14 (1922), pp. 55-62, 122-138 and 143-146.
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).

Muniru A Asiru, Table of n, a(n) for n = 2..300(Terms 2..170 from Vincenzo Librandi)
Anwar Al Ghabra, K. Gopala Krishna, Patrick Labelle, and Vasilisa Shramchenko, Enumeration of multi-rooted plane trees, arXiv:2301.09765 [math.CO], 2023. (Cites this sequence as "A00344")
Jean-Luc Baril and Helmut Prodinger, Enumeration of partial Lukasiewicz paths, arXiv:2205.01383 [math.CO], 2022.
Paul Barry, A Catalan Transform and Related Transformations on Integer Sequences, Journal of Integer Sequences, Vol. 8 (2005), Article 05.4.5.
Olivier Danvy, Summa Summarum: Moessner's Theorem without Dynamic Programming, arXiv:2412.03127 [cs.DM], 2024. See p. 31.
Dennis E. Davenport, Lara K. Pudwell, Louis W. Shapiro, and Leon C. Woodson, The Boundary of Ordered Trees, Journal of Integer Sequences, Vol. 18 (2015), Article 15.5.8.
Hilmar Haukur Gudmundsson, Dyck paths, standard Young tableaux, and pattern avoiding permutations, PU. M. A., Vol. 21, No. 2 (2010), pp. 265-284 (see Theorem 4.2 p. 275).
Richard K. Guy, Catwalks, sandsteps and Pascal pyramids, J. Integer Sequences, Vol. 3 (2000), Article #00.1.6.
V. E. Hoggatt, Jr. and M. Bicknell, Catalan and related sequences arising from inverses of Pascal's triangle matrices, Fib. Quart., Vol. 14, No. 5 (1976), pp. 395-405.
C. Krishnamachary and M. Bheemasena Rao, Determinants whose elements are Eulerian, prepared Bernoullian and other numbers, J. Indian Math. Soc., Vol. 14 (1922), pp. 55-62, 122-138 and 143-146. [Annotated scanned copy]
Athanasios Papoulis, A new method of inversion of the Laplace transform, Quart. Appl. Math., Vol. 14 (1957), pp. 405-414. [Annotated scan of selected pages]
Athanasios Papoulis, A new method of inversion of the Laplace transform, Quart. Applied Math., Vol. 14 (1956), pp. 405-414.
John Riordan, Letter to N. J. A. Sloane, Nov 10 1970.
John Riordan, The distribution of crossings of chords joining pairs of 2n points on a circle, Math. Comp., Vol. 29, No. 129 (1975), pp. 215-222.
Zoran Sunic, Self-Describing Sequences and the Catalan Family Tree, Electronic Journal of Combinatorics, Vol. 10 (2003) Article N5.

T(n, n+5) for n=0, 1, 2, ..., array T as in A047072.
A diagonal of any of the essentially equivalent arrays A009766, A030237, A033184, A059365, A099039, A106566, A130020, A047072.
Cf. A000108, A000245, A002057, A003517, A000588, A003518, A003519, A001392, A001622.

List([2..30],n->5*Binomial(2*n,n-2)/(n+3)); # Muniru A Asiru, Aug 09 2018

[5*Binomial(2*n,n-2)/(n+3): n in [2..30]]; // Vincenzo Librandi, May 03 2011

A000344List := proc(m) local A, P, n; A := [1]; P := [1,1,1,1];
for n from 1 to m - 2 do P := ListTools:-PartialSums([op(P), P[-1]]);
A := [op(A), P[-1]] od; A end: A000344List(27); # Peter Luschny, Mar 26 2022

Table[5 Binomial[2n,n-2]/(n+3),{n,2,40}] (* or *) CoefficientList[Series[ (1-Sqrt[1-4 x]+x (-5+3 Sqrt[1-4 x]-(-5+Sqrt[1-4 x]) x))/(2 x^5), {x,0,38}],x]  (* Harvey P. Dale, May 01 2011 *)
a[ n_] := If[ n < 0, 0, 5 Binomial[2 n, n - 2] / (n + 3)]; (* Michael Somos, May 28 2014 *)

a(n)=5*binomial(2*n,n-2)/(n+3) \\ Charles R Greathouse IV, Jul 25 2011

Integral representation as n-th moment of a function on [0, 4]: a(n) = Integral_{x=0..4} x^n*((1/2)/Pi*x^(3/2)*(x^2-3*x+1)*(4-x)^(1/2)) dx, n >= 0, for which offset=0. - Karol A. Penson, Oct 11 2001
Expansion of x^2*C^5, where C = (1-(1-4*x)^(1/2))/(2*x) is g.f. for Catalan numbers (A000108). - Herbert Kociemba, May 02 2004
Let A be the Toeplitz matrix of order n defined by: A[i,i-1]=-1, A[i,j]=Catalan(j-i), (i<=j), and A[i,j]=0, otherwise. Then, for n>=4, a(n-2)=(-1)^(n-4)*coeff(charpoly(A,x),x^4). - Milan Janjic, Jul 08 2010
a(n) = A000108(n+2) - 3*A000108(n+1)+ A000108(n). - David Scambler, May 20 2012
D-finite with recurrence: (n+3)*(n-2)*a(n) = 2*n*(2n-1)*a(n-1). - R. J. Mathar, Jun 27 2012
a(n) = A214292(2*n-1,n-3) for n > 2. - Reinhard Zumkeller, Jul 12 2012
0 = a(n)*(-528*a(n+1) + 9162*a(n+2) - 9295*a(n+3) + 1859*a(n+4)) + a(n+1)*(-1650*a(n+1) - 762*a(n+2) + 4188*a(n+3) - 946*a(n+4)) + a(n+2)*(-1050*a(n+2) - 126*a(n+3) + 84*a(n+4)) for all n in Z. - Michael Somos, May 28 2014
0 = a(n)*(a(n)*(+16*a(n+1) + 6*a(n+2)) + a(n+1)*(+66*a(n+1) - 105*a(n+2) + 40*a(n+3)) + a(n+2)*(-69*a(n+2) + 15*a(n+3))) +a(n+1)*(a(n+1)*(50*a(n+1) + 42*a(n+2) - 28*a(n+3)) +a(n+2)*(+12*a(n+2))) for all n in Z. - Michael Somos, May 28 2014
0 = a(n)^2*(-16*a(n+1)^2 - 38*a(n+1)*a(n+2) - 12*a(n+2)^2) + a(n)*a(n+1)*(-66*a(n+1)^2 + 149*a(n+1)*a(n+2) - 23*a(n+2)^2) + a(n+1)^2*(-50*a(n+1)^2 + 2*a(n+2)^2) for all n in Z. - Michael Somos, May 28 2014
From Ilya Gutkovskiy, Jan 22 2017: (Start)
E.g.f.: (x*(2 + x) * BesselI(0, 2*x) - (2+x+x^2) * BesselI(1, 2*x)) * exp(2*x)/x^2.
a(n) ~ 5*4^n/(sqrt(Pi)*n^(3/2)). (End)
a(n) = (1/(n+1))*Sum_{i=0..n-2} (-1)^(n+i)*(n-i+1)*binomial(2n+2,i), n >= 2. - Taras Goy, Aug 09 2018
G.f.: x^2* 2F1(5/2,3;6;4*x) . - R. J. Mathar, Jan 27 2020
From Amiram Eldar, Jan 02 2022: (Start)
Sum_{n>=2} 1/a(n) = 14/5 - 38*Pi/(45*sqrt(3)).
Sum_{n>=2} (-1)^n/a(n) = 1956*log(phi)/(125*sqrt(5)) - 316/125, where phi is the golden ratio (A001622). (End)
a(n) = 5*(2*n)!*(n-1)!/((2*n-4)!*(n+3)!)*A000108(n-2). - Taras Goy, Jul 15 2024
a(n) = Sum_{i+j+k+l+m = n-2} C(i)C(j)C(k)C(l)C(m), where C(s) = A000108(s). (Fifth convolution of Catalan numbers). - Taras Goy, Dec 21 2024

A228196 A triangle formed like Pascal's triangle, but with n^2 on the left border and 2^n on the right border instead of 1.

Original entry on oeis.org

0, 1, 2, 4, 3, 4, 9, 7, 7, 8, 16, 16, 14, 15, 16, 25, 32, 30, 29, 31, 32, 36, 57, 62, 59, 60, 63, 64, 49, 93, 119, 121, 119, 123, 127, 128, 64, 142, 212, 240, 240, 242, 250, 255, 256, 81, 206, 354, 452, 480, 482, 492, 505, 511, 512, 100, 287, 560, 806, 932, 962, 974, 997, 1016, 1023, 1024

Offset: 1

Boris Putievskiy, Aug 15 2013

The third row is (n^4 - n^2 + 24*n + 24)/12.

For a closed-form formula for generalized Pascal's triangle see A228576. - Boris Putievskiy, Sep 04 2013

			The start of the sequence as a triangular array read by rows:
   0;
   1,  2;
   4,  3,  4;
   9,  7,  7,  8;
  16, 16, 14, 15, 16;
  25, 32, 30, 29, 31, 32;
  36, 57, 62, 59, 60, 63, 64;

Boris Putievskiy, Rows n = 1..140 of triangle, flattened
Rely Pellicer and David Alvo, Modified Pascal Triangle and Pascal Surfaces p.4
Index entries for triangles and arrays related to Pascal's triangle

Cf. We denote Pascal-like triangle with L(n) on the left border and R(n) on the right border by (L(n),R(n)). A007318 (1,1), A008949 (1,2^n), A029600 (2,3), A029618 (3,2), A029635 (1,2), A029653 (2,1), A037027 (Fibonacci(n),1), A051601 (n,n) n>=0, A051597 (n,n) n>0, A051666 (n^2,n^2), A071919 (1,0), A074829 (Fibonacci(n), Fibonacci(n)), A074909 (1,n), A093560 (3,1), A093561 (4,1), A093562 (5,1), A093563 (6,1), A093564 (7,1), A093565 (8,1), A093644 (9,1), A093645 (10,1), A095660 (1,3), A095666 (1,4), A096940 (1,5), A096956 (1,6), A106516 (3^n,1), A108561(1,(-1)^n), A132200 (4,4), A134636 (2n+1,2n+1), A137688 (2^n,2^n), A160760 (3^(n-1),1), A164844(1,10^n), A164847 (100^n,1), A164855 (101*100^n,1), A164866 (101^n,1), A172171 (1,9), A172185 (9,11), A172283 (-9,11), A177954 (int(n/2),1), A193820 (1,2^n), A214292 (n,-n), A227074 (4^n,4^n), A227075 (3^n,3^n), A227076 (5^n,5^n), A227550 (n!,n!), A228053 ((-1)^n,(-1)^n), A228074 (Fibonacci(n), n).

Cf. A000290 (row 1), A153056 (row 2), A000079 (column 1), A000225 (column 2), A132753 (column 3), A118885 (row sums of triangle array + 1), A228576 (generalized Pascal's triangle).

T:= function(n,k)
    if k=0 then return n^2;
    elif k=n then return 2^n;
    else return T(n-1,k-1) + T(n-1,k);
    fi;
  end;
Flat(List([0..12], n-> List([0..n], k-> T(n,k) ))); # G. C. Greubel, Nov 12 2019

T:= proc(n, k) option remember;
      if k=0 then n^2
    elif k=n then 2^k
    else T(n-1, k-1) + T(n-1, k)
      fi
    end:
seq(seq(T(n, k), k=0..n), n=0..10); # G. C. Greubel, Nov 12 2019

T[n_, k_]:= T[n, k] = If[k==0, n^2, If[k==n, 2^k, T[n-1, k-1] + T[n-1, k]]]; Table[T[n, k], {n,0,10}, {k,0,n}]//Flatten (* G. C. Greubel, Nov 12 2019 *)
Flatten[Table[Sum[i^2 Binomial[n-1-i, n-k-i], {i,1,n-k}] + Sum[2^i Binomial[n-1-i, k-i], {i,1,k}], {n,0,10}, {k,0,n}]] (* Greg Dresden, Aug 06 2022 *)

T(n,k) = if(k==0, n^2, if(k==n, 2^k, T(n-1, k-1) + T(n-1, k) )); \\ G. C. Greubel, Nov 12 2019

def funcL(n):
   q = n**2
   return q
def funcR(n):
   q = 2**n
   return q
for n in range (1,9871):
   t=int((math.sqrt(8*n-7) - 1)/ 2)
   i=n-t*(t+1)/2-1
   j=(t*t+3*t+4)/2-n-1
   sum1=0
   sum2=0
   for m1 in range (1,i+1):
      sum1=sum1+funcR(m1)*binomial(i+j-m1-1,i-m1)
   for m2 in range (1,j+1):
      sum2=sum2+funcL(m2)*binomial(i+j-m2-1,j-m2)
   sum=sum1+sum2

@CachedFunction
def T(n, k):
    if (k==0): return n^2
    elif (k==n): return 2^n
    else: return T(n-1, k-1) + T(n-1, k)
[[T(n, k) for k in (0..n)] for n in (0..12)] # G. C. Greubel, Nov 12 2019

T(n,0) = n^2, n>0; T(0,k) = 2^k; T(n, k) = T(n-1, k-1) + T(n-1, k) for n,k > 0. [corrected by G. C. Greubel, Nov 12 2019]
Closed-form formula for general case. Let L(m) and R(m) be the left border and the right border of Pascal like triangle, respectively. We denote binomial(n,k) by C(n,k).
As table read by antidiagonals T(n,k) = Sum_{m1=1..n} R(m1)*C(n+k-m1-1, n-m1) + Sum_{m2=1..k} L(m2)*C(n+k-m2-1, k-m2); n,k >=0.
As linear sequence a(n) = Sum_{m1=1..i} R(m1)*C(i+j-m1-1, i-m1) + Sum_{m2=1..j} L(m2)*C(i+j-m2-1, j-m2), where i=n-t*(t+1)/2-1, j=(t*t+3*t+4)/2-n-1, t=floor((-1+sqrt(8*n-7))/2); n>0.
Some special cases. If L(m)={b,b,b...} b*A000012, then the second sum takes form b*C(n+k-1,j). If L(m) is {0,b,2b,...} b*A001477, then the second sum takes form b*C(n+k,n-1). Similarly for R(m) and the first sum.
For this sequence L(m)=m^2 and R(m)=2^m.
As table read by antidiagonals T(n,k) = Sum_{m1=1..n} (2^m1)*C(n+k-m1-1, n-m1) + Sum_{m2=1..k} (m2^2)*C(n+k-m2-1, k-m2); n,k >=0.
As linear sequence a(n) = Sum_{m1=1..i} (2^m1)*C(i+j-m1-1, i-m1) + Sum_{m2=1..j} (m2^2)*C(i+j-m2-1, j-m2), where i=n-t*(t+1)/2-1, j=(t*t+3*t+4)/2-n-1, t=floor((-1+sqrt(8*n-7))/2).
As a triangular array read by rows, T(n,k) = Sum_{i=1..n-k} i^2*C(n-1-i, n-k-i) + Sum_{i=1..k} 2^i*C(n-1-i, k-i); n,k >=0. - Greg Dresden, Aug 06 2022

Cross-references corrected and extended by Philippe Deléham, Dec 27 2013

A000588 a(n) = 7*binomial(2n,n-3)/(n+4).

Original entry on oeis.org

0, 0, 0, 1, 7, 35, 154, 637, 2548, 9996, 38760, 149226, 572033, 2187185, 8351070, 31865925, 121580760, 463991880, 1771605360, 6768687870, 25880277150, 99035193894, 379300783092, 1453986335186, 5578559816632, 21422369201800, 82336410323440, 316729578421620

Offset: 0

N. J. A. Sloane

a(n-5) is the number of n-th generation vertices in the tree of sequences with unit increase labeled by 6 (cf. Zoran Sunic reference). - Benoit Cloitre, Oct 07 2003
Number of lattice paths from (0,0) to (n,n) with steps E=(1,0) and N=(0,1) which touch but do not cross the line x-y=3. Example: For n=3 there is only one path EEENNN. - Herbert Kociemba, May 24 2004
Number of standard tableaux of shape (n+3,n-3). - Emeric Deutsch, May 30 2004

			G.f. = x^3 + 7*x^4 + 35*x^5 + 154*x^6 + 637*x^7 + 2548*x^8 + 9996*x^9 + ...

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
Joerg Arndt, The a(5)=35 Young tableaux of shape [8,2].
Dennis E. Davenport, Louis W. Shapiro, Lara K. Pudwell and Leon C. Woodson, The Boundary of Ordered Trees, J. Integer Seq., Vol. 18 (2015), Article 15.5.8; alternative link.
Hilmar Haukur Gudmundsson, Dyck paths, standard Young tableaux, and pattern avoiding permutations, PU. M. A., Vol. 21, No. 2 (2010), pp. 265-284 (see 4.4 p. 279).
Richard K. Guy, Catwalks, sandsteps and Pascal pyramids, J. Integer Sequences, Vol. 3 (2000), Article #00.1.6.
V. E. Hoggatt, Jr. and M. Bicknell, Catalan and related sequences arising from inverses of Pascal's triangle matrices, Fib. Quart., Vol. 14, No. 5 (1976), pp. 395-405.
Athanasios Papoulis, A new method of inversion of the Laplace transform, Quart. Appl. Math., Vol. 14 (1957), pp. 405-414. [Annotated scan of selected pages]
Athanasios Papoulis, A new method of inversion of the Laplace transform, Quart. Applied Math., Vol. 14 (1956), pp. 405-414.
John Riordan, Letter to N. J. A. Sloane, Nov 10 1970.
John Riordan, The distribution of crossings of chords joining pairs of 2n points on a circle, Math. Comp., Vol. 29, No. 129 (1975), pp. 215-222.
Zoran Sunic, Self-Describing Sequences and the Catalan Family Tree, Electronic Journal of Combinatorics, Vol. 10 (2003), Article N5.

First differences are in A026014.
A diagonal of any of the essentially equivalent arrays A009766, A030237, A033184, A059365, A099039, A106566, A130020, A047072.
Cf. A000108, A000245, A002057, A000344, A003517, A003518, A003519, A001392, A001622.

a[n_] := 7*Binomial[2n, n-3]/(n + 4); Table[a[n],{n,0,27}] (* James C. McMahon, Dec 05 2023 *)

A000588(n)=7*binomial(2*n,n-3)/(n+4) \\ M. F. Hasler, Aug 25 2012

my(x='x+O('x^50)); concat([0, 0, 0], Vec(x^3*((1-(1-4*x)^(1/2))/(2*x))^7)) \\ Altug Alkan, Nov 01 2015

Expansion of x^3*C^7, where C = (1-(1-4*x)^(1/2))/(2*x) is the g.f. for the Catalan numbers, A000108. - Philippe Deléham, Feb 03 2004
Let A be the Toeplitz matrix of order n defined by: A[i,i-1]=-1, A[i,j]=Catalan(j-i), (i<=j), and A[i,j]=0, otherwise. Then, for n>=6, a(n-3)=(-1)^(n-6)*coeff(charpoly(A,x),x^6). - Milan Janjic, Jul 08 2010
a(n) = A214292(2*n-1,n-4) for n > 3. - Reinhard Zumkeller, Jul 12 2012
From Ilya Gutkovskiy, Jan 22 2017: (Start)
E.g.f.: (1/6)*x^3*1F1(7/2; 8; 4*x).
a(n) ~ 7*4^n/(sqrt(Pi)*n^(3/2)). (End)
0 = a(n)*(+1456*a(n+1) - 87310*a(n+2) + 132834*a(n+3) - 68068*a(n+4) + 9724*a(n+5)) + a(n+1)*(+8918*a(n+1) - 39623*a(n+2) + 51726*a(n+3) - 299*a(n+4) - 1573*a(n+5)) + a(n+2)*(-24696*a(n+2) - 1512*a(n+3) + 1008*a(n+4)) for all n in Z. - Michael Somos, Jan 22 2017
From Amiram Eldar, Jan 02 2022: (Start)
Sum_{n>=3} 1/a(n) = 27/14 - 26*Pi/(63*sqrt(3)).
Sum_{n>=3} (-1)^(n+1)/a(n) = 11364*log(phi)/(175*sqrt(5)) - 4583/350, where phi is the golden ratio (A001622). (End)
a(n) = Integral_{x=0..4} x^(n)*W(x)dx, n>=0, where W(x) = sqrt(4/x - 1)*(x^3 - 5*x^2 + 6*x - 1)/(2*Pi). The function W(x) for x->0 tends to -infinity (which is its absolute minimum), and W(4) = 0. W(x) is a signed function on the interval x = (0, 4) where it has two maxima separated by one local minimum. - Karol A. Penson, Jun 17 2024
D-finite with recurrence -(n+4)*(n-3)*a(n) +2*n*(2*n-1)*a(n-1)=0. - R. J. Mathar, Jul 30 2024
a(n) = A000108(n+3) - 5*A000108(n+2) + 6*A000108(n+1) - A000108(n). - Taras Goy, Dec 21 2024

More terms from N. J. A. Sloane, Jul 13 2010

A001392 a(n) = 9*binomial(2n,n-4)/(n+5).

Original entry on oeis.org

1, 9, 54, 273, 1260, 5508, 23256, 95931, 389367, 1562275, 6216210, 24582285, 96768360, 379629720, 1485507600, 5801732460, 22626756594, 88152205554, 343176898988, 1335293573130, 5193831553416, 20198233818840, 78542105700240, 305417807763705

Offset: 4

N. J. A. Sloane

Number of n-th generation vertices in the tree of sequences with unit increase labeled by 8 (cf. Zoran Sunic reference) - Benoit Cloitre, Oct 07 2003
Number of lattice paths from (0,0) to (n,n) with steps E=(1,0) and N=(0,1) which touch but do not cross the line x-y=4. - Herbert Kociemba, May 24 2004
Number of standard tableaux of shape (n+4,n-4). - Emeric Deutsch, May 30 2004

			G.f. = x^4 + 9*x^5 + 54*x^6 + 273*x^7 + 1260*x^8 + 5508*x^9 + 23256*x^10 + ...

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 = 4..200
Richard K. Guy, Catwalks, sandsteps and Pascal pyramids, J. Integer Sequences, Vol. 3 (2000), Article #00.1.6.
Athanasios Papoulis, A new method of inversion of the Laplace transform, Quart. Appl. Math., Vol. 14 (1957), pp. 405-414. [Annotated scan of selected pages]
Athanasios Papoulis, A new method of inversion of the Laplace transform, Quart. Applied Math., Vol. 14 (1957), pp. 405-414.
John Riordan, The distribution of crossings of chords joining pairs of 2n points on a circle, Math. Comp., Vol. 29, No. 129 (1975), pp. 215-222.
Zoran Sunic, Self-Describing Sequences and the Catalan Family Tree, Electronic Journal of Combinatorics, Vol. 10 (2003), Article #N5.

First differences are in A026015.
A diagonal of any of the essentially equivalent arrays A009766, A030237, A033184, A059365, A099039, A106566, A130020, A047072.
Cf. A000108, A000245, A002057, A000344, A003517, A000588, A003518, A003519, A001622.

A001392:=n->9*binomial(2*n,n-4)/(n+5): seq(A001392(n), n=4..40); # Wesley Ivan Hurt, Apr 11 2017

Table[9*Binomial[2n,n-4]/(n+5),{n,4,30}] (* Harvey P. Dale, Mar 03 2011 *)

a(n)=9*binomial(n+n,n-4)/(n+5) \\ Charles R Greathouse IV, Jul 31 2011

Expansion of x^4*C^9, where C = (1-(1-4*x)^(1/2))/(2*x) is g.f. for Catalan numbers, A000108. - Philippe Deléham, Feb 03 2004
Let A be the Toeplitz matrix of order n defined by: A[i,i-1]=-1, A[i,j]=Catalan(j-i), (i<=j), and A[i,j]=0, otherwise. Then, for n>=8, a(n-4)=(-1)^(n-8)*coeff(charpoly(A,x),x^8). - Milan Janjic, Jul 08 2010
a(n) = A214292(2*n-1,n-5) for n > 4. - Reinhard Zumkeller, Jul 12 2012
D-finite with recurrence -(n+5)*(n-4)*a(n) +2*n*(2*n-1)*a(n-1)=0. - R. J. Mathar, Jun 20 2013
From Ilya Gutkovskiy, Jan 22 2017: (Start)
E.g.f.: (1/24)*x^4*1F1(9/2; 10; 4*x).
a(n) ~ 9*4^n/(sqrt(Pi)*n^(3/2)). (End)
From Amiram Eldar, Jan 02 2022: (Start)
Sum_{n>=4} 1/a(n) = 158*Pi/(81*sqrt(3)) - 649/270.
Sum_{n>=4} (-1)^n/a(n) = 52076*log(phi)/(225*sqrt(5)) - 22007/450, where phi is the golden ratio (A001622). (End)

More terms from Harvey P. Dale, Mar 03 2011

A003518 a(n) = 8binomial(2n+1,n-3)/(n+5).

Original entry on oeis.org

1, 8, 44, 208, 910, 3808, 15504, 62016, 245157, 961400, 3749460, 14567280, 56448210, 218349120, 843621600, 3257112960, 12570420330, 48507033744, 187187399448, 722477682080, 2789279908316, 10772391370048, 41620603020640, 160878516023680, 622147386185325

Offset: 3

N. J. A. Sloane

a(n-6) is the number of n-th generation nodes in the tree of sequences with unit increase labeled by 7 (cf. Zoran Sunic reference). - Benoit Cloitre, Oct 07 2003

Number of standard tableaux of shape (n+4,n-3). - Emeric Deutsch, May 30 2004

			G.f. = x^3 + 8*x^4 + 44*x^5 + 208*x^6 + 910*x^7 + 3808*x^8 + 15504*x^9 + ...

N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Vincenzo Librandi, Table of n, a(n) for n = 3..500
Daniel Birmajer, Juan B. Gil and Michael D. Weiner, Bounce statistics for rational lattice paths, arXiv:1707.09918 [math.CO], 2017, p. 9.
S. J. Cyvin, J. Brunvoll, E. Brendsdal, B. N. Cyvin and E. K. Lloyd, Enumeration of polyene hydrocarbons: a complete mathematical solution, J. Chem. Inf. Comput. Sci., Vol. 35, No. 4 (1995), pp. 743-751.
S. J. Cyvin, J. Brunvoll, E. Brendsdal, B. N. Cyvin and E. K. Lloyd, Enumeration of polyene hydrocarbons: a complete mathematical solution, J. Chem. Inf. Comput. Sci., Vol. 35, No. 4 (1995), pp. 743-751. [Annotated scanned copy]
Hilmar Haukur Gudmundsson, Dyck paths, standard Young tableaux, and pattern avoiding permutations, PU. M. A., Vol. 21, No.2 (2010), pp. 265-284; arXiv:0912.4747 [math.CO], 2009 (see Theorem 11 in Section 4.5).
Richard K. Guy, Letter to N. J. A. Sloane, May 1990.
Richard K. Guy, Catwalks, Sandsteps and Pascal Pyramids, J. Integer Seq., Vol. 3 (2000), Article 00.1.6.
V. E. Hoggatt, Jr., 7-page typed letter to N. J. A. Sloane with suggestions for new sequences, circa 1977.
V. E. Hoggatt, Jr. and M. Bicknell, Catalan and related sequences arising from inverses of Pascal's triangle matrices, Fib. Quart., Vol. 14, No. 5 (1976), pp. 395-405.
Olya Mandelshtam, Multi-Catalan tableaux and the two-species TASEP, arXiv:1502.00948 [math.CO], 2015.
Olya Mandelshtam, Multi-Catalan tableaux and the two-species TASEP, Ann. Inst. Henri Poincaré Comb. Phys. Interact., Vol. 3 (2016), pp. 321-348, DOI 10.4171/AIHPD/30.
L. W. Shapiro, A Catalan triangle, Discrete Math., Vol. 14, No. 1 (1976), pp. 83-90.
L. W. Shapiro, A Catalan triangle, Discrete Math., Vol. 14, No. 1 (1976), pp. 83-90. [Annotated scanned copy]
Zoran Sunic, Self describing sequences and the Catalan family tree, Elect. J. Combin., Vol. 10 (2003), Article N5.
Wen-Jin Woan, Lou Shapiro and D. G. Rogers, The Catalan numbers, the Lebesgue integral and 4^{n-2}, Amer. Math. Monthly, Vol. 104, No. 10 (1997), pp. 926-931.

Cf. A002057.
First differences are in A026018.
A diagonal of any of the essentially equivalent arrays A009766, A030237, A033184, A059365, A099039, A106566, A130020, A047072.
Cf. A000108, A000245, A000344, A000588, A001392, A002057, A003517, A003519, A001622.

[8*Binomial(2*n+1,n-3)/(n+5): n in [3..30]]; // Vincenzo Librandi, Jan 23 2017

Table[8 Binomial[2 n + 1, n - 3]/(n + 5), {n, 3, 25}] (* Michael De Vlieger, Oct 26 2016 *)
CoefficientList[Series[((1 - Sqrt[1 - 4 x])/(2 x))^8, {x, 0, 30}], x] (* Vincenzo Librandi, Jan 23 2017 *)

{a(n) = if( n<3, 0, 8 * binomial(2*n + 1, n-3) / (n + 5))}; /* Michael Somos, Mar 14 2011 */

my(x='x+O('x^50)); Vec(x^3*((1-(1-4*x)^(1/2))/(2*x))^8) \\ Altug Alkan, Nov 01 2015

G.f.: x^3*C(x)^8, where C(x)=(1-sqrt(1-4*x))/(2*x) is g.f. for the Catalan numbers (A000108). - Emeric Deutsch, May 30 2004
The convolution of A002057 with itself. - Gerald McGarvey, Nov 08 2007
Let A be the Toeplitz matrix of order n defined by: A[i,i-1]=-1, A[i,j]=Catalan(j-i), (i<=j), and A[i,j]=0, otherwise. Then, for n>=7, a(n-4)=(-1)^(n-7)*coeff(charpoly(A,x),x^7). - Milan Janjic, Jul 08 2010
a(n) = A214292(2*n,n-4) for n > 3. - Reinhard Zumkeller, Jul 12 2012
Integral representation as the n-th moment of the signed weight function W(x) on (0,4), i.e.: a(n+3) = Integral_{x=0..4} x^n*W(x) dx, n >= 0, with W(x) = (1/2)*x^(7/2)*(x-2)*(x^2-4*x+2)*sqrt(4-x)/Pi. - Karol A. Penson, Oct 26 2016
From Ilya Gutkovskiy, Jan 22 2017: (Start)
E.g.f.: 4*BesselI(4,2*x)*exp(2*x)/x.
a(n) ~ 4^(n+2)/(sqrt(Pi)*n^(3/2)). (End)
D-finite with recurrence: -(n+5)*(n-3)*a(n) +2*n*(2*n+1)*a(n-1)=0. - R. J. Mathar, Feb 20 2020
From Amiram Eldar, Jan 02 2022: (Start)
Sum_{n>=3} 1/a(n) = 43*Pi/(36*sqrt(3)) - 81/80.
Sum_{n>=3} (-1)^(n+1)/a(n) = 6213*log(phi)/(50*sqrt(5)) - 10339/400, where phi is the golden ratio (A001622). (End)

More terms from Jon E. Schoenfield, May 06 2010

A003519 a(n) = 10*C(2n+1, n-4)/(n+6).

Original entry on oeis.org

1, 10, 65, 350, 1700, 7752, 33915, 144210, 600875, 2466750, 10015005, 40320150, 161280600, 641886000, 2544619500, 10056336264, 39645171810, 155989499540, 612815891050, 2404551645100, 9425842448792, 36921502679600, 144539291740025, 565588532895750, 2212449261033375

Offset: 4

N. J. A. Sloane

Number of standard tableaux of shape (n+5,n-4). - Emeric Deutsch, May 30 2004

a(n) is the number of North-East paths from (0,0) to (n,n) that cross the diagonal y = x horizontally exactly twice. By symmetry, it is also the number of North-East paths from (0,0) to (n,n) that cross the diagonal y = x vertically exactly twice. Details can be found in Section 3.3 in Pan and Remmel's link. - Ran Pan, Feb 02 2016

N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Robert Israel, Table of n, a(n) for n = 4..1650
Daniel Birmajer, Juan B. Gil and Michael D. Weiner, Bounce statistics for rational lattice paths, arXiv:1707.09918 [math.CO], 2017, p. 9.
Richard K. Guy, Letter to N. J. A. Sloane, May 1990.
Richard K. Guy, Catwalks, sandsteps and Pascal pyramids, J. Integer Sequences, Vol. 3 (2000), Article #00.1.6.
V. E. Hoggatt, Jr., 7-page typed letter to N. J. A. Sloane with suggestions for new sequences, circa 1977.
V. E. Hoggatt, Jr. and M. Bicknell, Catalan and related sequences arising from inverses of Pascal's triangle matrices, Fib. Quart., Vol. 14, No. 5 (1976), pp. 395-405.
Ran Pan and Jeffrey B. Remmel, Paired patterns in lattice paths, arXiv:1601.07988 [math.CO], 2016.

A diagonal of any of the essentially equivalent arrays A009766, A030237, A033184, A059365, A099039, A106566, A130020, A047072.

Cf. A000108, A000245, A002057, A000344, A003517, A000588, A003518, A001392, A001622.

[10*Binomial(2*n+1, n-4)/(n+6): n in [4..35]]; // Vincenzo Librandi, Feb 03 2016

seq(10*binomial(2*n+1,n-4)/(n+6), n=4..50); # Robert Israel, Feb 02 2016

Table[10 Binomial[2 n + 1, n - 4]/(n + 6), {n, 4, 28}] (* Michael De Vlieger, Feb 03 2016 *)

a(n) = 10*binomial(2*n+1, n-4)/(n+6); \\ Michel Marcus, Feb 02 2016

G.f.: x^4*C(x)^10, where C(x)=[1-sqrt(1-4x)]/(2x) is g.f. for the Catalan numbers (A000108). - Emeric Deutsch, May 30 2004
Let A be the Toeplitz matrix of order n defined by: A[i,i-1]=-1, A[i,j]=Catalan(j-i), (i<=j), and A[i,j]=0, otherwise. Then, for n>=9, a(n-5)=(-1)^(n-9)*coeff(charpoly(A,x),x^9). [Milan Janjic, Jul 08 2010]
a(n) = A214292(2*n,n-5) for n > 4. - Reinhard Zumkeller, Jul 12 2012
From Robert Israel, Feb 02 2016: (Start)
D-finite with recurrence a(n+1) = 2*(n+1)*(2n+3)/((n+7)*(n-3)) * a(n).
a(n) ~ 20 * 4^n/sqrt(Pi*n^3). (End)
E.g.f.: 5*BesselI(5,2*x)*exp(2*x)/x. - Ilya Gutkovskiy, Jan 23 2017
From Amiram Eldar, Jan 02 2022: (Start)
Sum_{n>=4} 1/a(n) = 34*Pi/(45*sqrt(3)) - 44/175.
Sum_{n>=4} (-1)^n/a(n) = 53004*log(phi)/(125*sqrt(5)) - 79048/875, where phi is the golden ratio (A001622). (End)

A005586 a(n) = n(n+4)(n+5)/6.

Original entry on oeis.org

0, 5, 14, 28, 48, 75, 110, 154, 208, 273, 350, 440, 544, 663, 798, 950, 1120, 1309, 1518, 1748, 2000, 2275, 2574, 2898, 3248, 3625, 4030, 4464, 4928, 5423, 5950, 6510, 7104, 7733, 8398, 9100, 9840, 10619, 11438, 12298, 13200, 14145, 15134, 16168, 17248

Offset: 0

N. J. A. Sloane

Number of walks on square lattice.
Number of standard tableaux of shape (n+2,3) (n >= 1). - Emeric Deutsch, May 20 2004
Number of left factors of Dyck paths from (0,0) to (n+5,n-1). E.g. a(1)=5 because we have UDUDUD, UDUUDD, UUDDUD, UUDUDD and UUUDDD, where U=(1,1) and D=(1,-1). - Emeric Deutsch, Jan 25 2005
Column 4 of Catalan triangle A009766. - Zerinvary Lajos, Nov 25 2006
Sum of first n triangular numbers minus next triangular number. - Vladimir Joseph Stephan Orlovsky, Oct 13 2009
Number of packed increasing tableaux of shape 3 X (n+1) with alphabet [n+4]. - Oliver Pechenik, Jan 03 2022

			G.f. = 5*x + 14*x^2 + 28*x^3 + 48*x^4 + 75*x^5 + 110*x^6 + 154*x^7 + ...

M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards Applied Math. Series 55, 1964 (and various reprintings), p. 796.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
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].
C. Gaetz, O. Pechenik, J. Striker, and J. P. Swanson, Curious cyclic sieving on increasing tableaux, arXiv:2112.09228 [math.CO], 2021. See Proposition 1.1 at the top of page 2.
Richard K. Guy, Letter to N. J. A. Sloane, Feb 1988
Richard K. Guy, Catwalks, Sandsteps and Pascal Pyramids, J. Integer Seqs., Vol. 3 (2000), Article 00.1.6.
Ângela Mestre and José Agapito, A Family of Riordan Group Automorphisms, J. Int. Seq., Vol. 22 (2019), Article 19.8.5.
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 (4,-6,4,-1).

Cf. A000217, A000292, A005581, A009766, A053121.

a(n)=A053121(n+5,n-1).

[n*(n+4)*(n+5)/6: n in [0..50]]; // Vincenzo Librandi, Jun 09 2013

[seq(binomial(n,3 )-binomial(n,1),n=4..48)]; # Zerinvary Lajos, Nov 25 2006
a:=n->sum ((j-3)*j/2,j=0..n): seq(a(n),n=4..48); # Zerinvary Lajos, Dec 17 2006
A005586:=z*(5-6*z+2*z**2)/(z-1)**4; # conjectured by Simon Plouffe in his 1992 dissertation
seq(sum(binomial(n,m), m=1..3)-n^2,n=5..49); # Zerinvary Lajos, Jun 19 2008

Clear[lst,n,a,f]; f[n_]:=n*(n+1)/2; a=0;lst={};Do[a+=f[n];AppendTo[lst,a-f[n+1]],{n,5!}];lst (* Vladimir Joseph Stephan Orlovsky, Oct 13 2009 *)
CoefficientList[Series[x (5 - 6 x + 2 x^2) / (1 - x)^4, {x, 0, 50}], x] (* Vincenzo Librandi, Jun 09 2013 *)
Table[(n(n+4)(n+5))/6,{n,0,50}] (* or *) LinearRecurrence[{4,-6,4,-1},{0,5,14,28},50] (* Harvey P. Dale, Jul 14 2018 *)

{a(n) = n * (n+4) * (n+5) / 6}; /* Michael Somos, Apr 13 2007 */

G.f.: x * (5 - 6*x + 2*x^2) / (1 - x)^4.
E.g.f.: (5*x + 2*x^2 + x^3/6) * exp(x). - Michael Somos, Apr 13 2007
Let t(n) = n*(n+1)/2, te(n) = (n+1)*(n+2)*(n+3)/6. Then a(n-4) = -2*t(n) + te(n-1), e.g., a(2) = -2*t(6) + te(5) = -2*21 + 56 = 14, where te(n) are the tetrahedral numbers A000292 and t(n) are the triangular numbers A000217. - Jon Perry, Jul 23 2003
a(n) = C(5+n, 3)-C(5+n, 2). - Zerinvary Lajos, Jan 09 2006
a(n) = C(n,3) - C(n,1), n>=4. - Zerinvary Lajos, Nov 25 2006
a(n) = - A005581(-4-n) for all n in Z. - Michael Somos, Apr 13 2007
a(n) = A214292(n+4,2). - Reinhard Zumkeller, Jul 12 2012
From Amiram Eldar, Feb 20 2022: (Start)
Sum_{n>=1} 1/a(n) = 77/200.
Sum_{n>=1} (-1)^(n+1)/a(n) = 363/200 - 12*log(2)/5. (End)
a(n) = A005581(n+2)-2. - R. J. Mathar, Nov 22 2024

M3842=A005555 in the 1995 EIS was the same sequence as this.

More terms from Zerinvary Lajos, Jan 09 2006

A014495 Central binomial coefficient - 1.

Original entry on oeis.org

0, 0, 1, 2, 5, 9, 19, 34, 69, 125, 251, 461, 923, 1715, 3431, 6434, 12869, 24309, 48619, 92377, 184755, 352715, 705431, 1352077, 2704155, 5200299, 10400599, 20058299, 40116599, 77558759, 155117519, 300540194, 601080389, 1166803109, 2333606219, 4537567649

Offset: 0

N. J. A. Sloane, Denis Pochuev (denis(AT)cdc.Informatik.TH-Darmstadt.de)

For n > 0: sum of positive elements in row (n-1) of triangle A214292. - Reinhard Zumkeller, Jul 12 2012
Number of Young tableaux with n cells and 2 rows.  Also number of self-inverse permutations in S_n with longest increasing subsequence of length 2.  The a(4) = 5 permutations are 1432, 2143, 3214, 3412, 4231 and the a(5) = 9 permutations are 15432, 21543, 32154, 35142, 42513, 43215, 45312, 52431, 53241. - Alois P. Heinz, Oct 03 2012
Number of nonempty subsets of {1,2,...,n} that contain the same number of even and odd numbers. For example, a(5)=9 and the 9 subsets are {1,2}, {1,4}, {2,3}, {2,5}, {3,4}, {4,5}, {1,2,3,4}, {1,2,4,5}, {2,3,4,5}. - Enrique Navarrete, Feb 10 2018

Alois P. Heinz, Table of n, a(n) for n = 0..1000

Cf. A001405, A037952 (first differences).
a(n) = A094718(n, n) = A094718(n-1, n)+1.
a(n) = A047884(n, 2) for n>=2. - Alois P. Heinz, Oct 03 2012
Cf. A214292.

[Binomial(n, Floor(n/2))-1: n in [0..50]]; // Vincenzo Librandi, Feb 11 2018

a:= n-> binomial(n, iquo(n, 2))-1:
seq(a(n), n=0..40);  # Alois P. Heinz, Oct 03 2012

Table[Binomial[n, Floor[n/2]] - 1, {n, 0, 50}] (* Bruno Berselli, Oct 03 2012 *)

A014495(n):=binomial(n,floor(n/2))-1$
makelist(A014495(n),n,0,30); /* Martin Ettl, Nov 01 2012 */

a(n) = A001405(n)-1.
a(n) = C(n,floor(n/2))-1. - Alois P. Heinz, Oct 03 2012
(n+1)*a(n)-2*a(n-1)-4*(n-1)*a(n-2) = 3*n-3 with n>1, a(0)=a(1)=0. - Bruno Berselli, Oct 03 2012
D-finite with recurrence: -(n+1)*(n-2)*a(n) +(n^2+n-4)*a(n-1) +2*(n-1)*(2*n-5)*a(n-2) -4*(n-1)*(n-2)*a(n-3)=0. - Conjectured by R. J. Mathar, Jan 04 2017, confirmed by Robert Israel, Feb 11 2018
G.f.: (x+1)/(2*x*(x-1)) - sqrt(1-4*x^2)/(2*x*(2*x-1)). - Robert Israel, Feb 11 2018

Edited by Andrey Zabolotskiy, Apr 14 2025

A000589 a(n) = 11*binomial(2n,n-5)/(n+6).

Original entry on oeis.org

1, 11, 77, 440, 2244, 10659, 48279, 211508, 904475, 3798795, 15737865, 64512240, 262256280, 1059111900, 4254603804, 17018415216, 67837293986, 269638992062, 1069258071970, 4232010895376, 16723268860760, 65997186039785, 260170725132045, 1024713341952300

Offset: 5

N. J. A. Sloane

nonn

Number of lattice paths from (0,0) to (n,n) with steps E=(1,0) and N=(0,1) which touch but do not cross the line x-y=5. - Herbert Kociemba, May 24 2004

Number of standard tableaux of shape (n+5,n-5). - Emeric Deutsch, May 30 2004

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 = 5..200
Richard K. Guy, Catwalks, sandsteps and Pascal pyramids, J. Integer Sequences, Vol. 3 (2000), Article 00.1.6.
Athanasios Papoulis, A new method of inversion of the Laplace transform, Quart. Applied Math. 14 (1956), 405ff.
Athanasios Papoulis, A new method of inversion of the Laplace transform, Quart. Appl. Math 14 (1957), 405-414. [Annotated scan of selected pages]
John Riordan, The distribution of crossings of chords joining pairs of 2n points on a circle, Math. Comp., 29 (1975), 215-222.

Cf. A000108, A001622, A214292.

a[n_] := 11*Binomial[2*n, n-5]/(n+6); Array[a, 25, 5] (* Amiram Eldar, Sep 26 2022 *)

Expansion of x^5*C^11, where C = (1-(1-4*x)^(1/2))/(2*x) is the g.f. for the Catalan numbers, A000108. - Philippe Deléham, Feb 03 2004
Let A be the Toeplitz matrix of order n defined by: A[i,i-1]=-1, A[i,j]=Catalan(j-i), (i<=j), and A[i,j]=0, otherwise. Then, for n>=10, a(n-5)=(-1)^(n-10)*coeff(charpoly(A,x),x^10). - Milan Janjic, Jul 08 2010
a(n) = A214292(2*n-1,n-6) for n > 5. - Reinhard Zumkeller, Jul 12 2012
-(n+6)*(n-5)*a(n) + 2*n*(2*n-1)*a(n-1) = 0. - R. J. Mathar, Jun 20 2013
From Amiram Eldar, Sep 26 2022: (Start)
Sum_{n>=5} 1/a(n) = 5993/1540 - 152*Pi/(99*sqrt(3)).
Sum_{n>=5} (-1)^(n+1)/a(n) = 210624*log(phi)/(275*sqrt(5)) - 1262077/7700, where phi is the golden ratio (A001622). (End)

A005587 a(n) = n(n+5)(n+6)*(n+7)/24.

Original entry on oeis.org

0, 14, 42, 90, 165, 275, 429, 637, 910, 1260, 1700, 2244, 2907, 3705, 4655, 5775, 7084, 8602, 10350, 12350, 14625, 17199, 20097, 23345, 26970, 31000, 35464, 40392, 45815, 51765, 58275, 65379, 73112, 81510, 90610, 100450, 111069, 122507, 134805, 148005

Offset: 0

N. J. A. Sloane

a(n) = number of Standard Young Tableaux of shape (n+3,4). - David Callan, Aug 17 2004
a(n) = A214292(n+6,3). - Reinhard Zumkeller, Jul 12 2012
a(n) for n > 0 is the number of n-extended coalescent histories for a matching caterpillar gene tree and species tree with 5 leaves. - Noah A Rosenberg, Jun 16 2022

M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards Applied Math. Series 55, 1964 (and various reprintings), p. 796.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
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].
Richard K. Guy, Letter to N. J. A. Sloane, Feb 1988.
Richard K. Guy, Catwalks, sandsteps and Pascal pyramids, J. Integer Sequences, Vol. 3 (2000), Article #00.1.6.
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.
N. A. Rosenberg, Counting coalescent histories, J. Comput. Biol. 14 (2007), 360-377.
Index entries for linear recurrences with constant coefficients, signature (5,-10,10,-5,1).

Fifth diagonal of Catalan triangle A033184. Fifth column of Catalan triangle A009766.

Numerator polynomial 14 - 28x + 20x^2 - 5x^3 from fourth row of triangle A062991.

[n*(n+5)*(n+6)*(n+7)/24: n in [0..40]]; // Vincenzo Librandi, Mar 20 2013

A005587:=z*(-14+28*z-20*z**2+5*z**3)/(z-1)**5; # Simon Plouffe in his 1992 dissertation
seq(numbperm(n,4)/24-numbperm(n,3)/6, n=7..46); # Zerinvary Lajos, May 20 2008
a:=n->(sum(numbcomp(n,4), j=9..n)):seq(a(n)/4, n=8..47); # Zerinvary Lajos, Aug 26 2008

Table[n (n + 5) (n + 6) (n + 7)/24, {n, 0, 60}] (* Vladimir Joseph Stephan Orlovsky, Jun 22 2011 *)
LinearRecurrence[{5,-10,10,-5,1},{0,14,42,90,165},40] (* Harvey P. Dale, Aug 17 2017 *)

x='x+O('x^50); concat([0], Vec((14 - 28*x + 20*x^2 - 5*x^3) / (1 - x)^5)) \\ G. C. Greubel, Jul 01 2017

G.f.: (14 - 28*x + 20*x^2 - 5*x^3) / (1 - x)^5.
a(n) = C(7+n, 4) - C(7+n, 3). - Zerinvary Lajos, Dec 09 2005
E.g.f.: (1/24)*x*(336 + 168*x + 24*x^2 + x^3)*exp(x). - G. C. Greubel, Jul 01 2017
From Amiram Eldar, Jun 28 2022: (Start)
Sum_{n>=1} 1/a(n) = 153/1225.
Sum_{n>=1} (-1)^(n+1)/a(n) = 288*log(2)/35 - 20759/3675. (End)
a(n) = A024191(n+1)-5. - R. J. Mathar, Nov 22 2024

M4929 (this sequence) and M4930 were the same.
More terms from Matthew Conroy, Jan 16 2006
Plouffe Maple line edited by N. J. A. Sloane, May 13 2008

cp's OEIS Frontend

A000344 a(n) = 5*binomial(2n, n-2)/(n+3).

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A228196 A triangle formed like Pascal's triangle, but with n^2 on the left border and 2^n on the right border instead of 1.

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A000588 a(n) = 7*binomial(2n,n-3)/(n+4).

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A001392 a(n) = 9*binomial(2n,n-4)/(n+5).

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A003518 a(n) = 8*binomial(2*n+1,n-3)/(n+5).

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A003518 a(n) = 8binomial(2n+1,n-3)/(n+5).

A005586 a(n) = n(n+4)(n+5)/6.

A005587 a(n) = n(n+5)(n+6)*(n+7)/24.