A033536 -id:A033536 - cp's OEIS frontend

A014137 Partial sums of Catalan numbers (A000108).

1, 2, 4, 9, 23, 65, 197, 626, 2056, 6918, 23714, 82500, 290512, 1033412, 3707852, 13402697, 48760367, 178405157, 656043857, 2423307047, 8987427467, 33453694487, 124936258127, 467995871777, 1757900019101, 6619846420553, 24987199492705, 94520750408709, 358268702159069

Offset: 0

N. J. A. Sloane

This is also the result of applying the transformation on generating functions A -> 1/((1 - x)*(1 - x*A)) to the g.f. for the Catalan numbers.
p divides a(p) - 3 for prime p = 3 and p = {7, 13, 19, 31, 37, 43, ...} = A002476 (Primes of the form 6*n + 1). p^2 divides a(p^2) - 3 for prime p > 3. - Alexander Adamchuk, Jul 11 2006
Prime p divides a(p) for p = {2, 3, 5, 11, 17, 23, 29, 41, 47, 53, 59, 71, 83, 89, 101, ...} = A045309 (Primes congruent to {0, 2} mod 3); and A045309 (Primes p such that x^3 = n (integer) has only one solution mod p). Nonprime numbers n such that n divides a(n) are listed in A128287 = {1, 8, 133, ...}. - Alexander Adamchuk, Feb 23 2007
For p prime >= 5, a(p-1) = 1 or -2 (mod p) according as p = 1 or -1 (mod 3) (see Pan and Sun link). For example, with p=5, a(p-1) = 23 = -2 (mod p). - David Callan, Nov 29 2007
Hankel transform is A010892(n+1). - Paul Barry, Apr 24 2009
Equals INVERTi transform of A000245: (1, 3, 9, 28, ...). - Gary W. Adamson, May 15 2009
The subsequence of prime partial sums of Catalan numbers begins: a(1) = 2, a(4) = 23, a(6) = 197, a(16) = 48760367; see A121852. - Jonathan Vos Post, Feb 10 2010
Number of lattice paths from (0,0) to (n,n) which do not go above the diagonal x=y using steps (1,k), (k,1) with k >= 1 including two kinds of (1,1). - Alois P. Heinz, Oct 14 2015
Binomial transform of A086246(n+1) = [1, 1, 1, 2, 4, 9, ...], or, equivalently, of A001006 (Motzkin numbers) with 1 prepended.

			G.f. = 1 + 2*x + 4*x^2 + 9*x^3 + 23*x^4 + 65*x^5 + 197*x^6 + 626*x^7 + 2056*x^8 + ...

G. C. Greubel, Table of n, a(n) for n = 0..1000 (terms 0 to 200 by T. D. Noe)
G. Alvarez, J. E. Bergner, and R. Lopez, Action Graphs and Catalan Numbers, J. Int. Seq. 18 (2015), 15.7.2.
M. Apagodu, D. Zeilberger, Using the Freshman's dream to prove combinatorial congruences, Am. Math. Mon. 124, No. 7, 597-608 (2017), prop. 2'
Maciej Bendkowski and Pierre Lescanne, Combinatorics of explicit substitutions, arXiv:1804.03862 [cs.LO], 2018.
W. Chammam, F. Marcellán and R. Sfaxi, Orthogonal polynomials, Catalan numbers, and a general Hankel determinant evaluation, Linear Algebra Appl. 436(7) (2012), 2105-2116.
Joel E. Cohen, Variance Functions of Asymptotically Exponentially Increasing Integer Sequences Go Beyond Taylor's Law, J. Int. Seq., Vol. 25 (2022), Article 22.9.3.
Dennis E. Davenport, Lara K. Pudwell, Louis W. Shapiro, and Leon C. Woodson, The Boundary of Ordered Trees, Journal of Integer Sequences, 18 (2015), Article 15.5.8.
Nancy S. S. Gu, Nelson Y. Li, and Toufik Mansour, 2-Binary trees: bijections and related issues, Discr. Math., 308 (2008), 1209-1221.
Guo-Niu Han, Enumeration of Standard Puzzles, 2011. [Cached copy]
Guo-Niu Han, Enumeration of Standard Puzzles, arXiv:2006.14070 [math.CO], 2020.
Ângela Mestre and José Agapito, A Family of Riordan Group Automorphisms, J. Int. Seq., Vol. 22 (2019), Article 19.8.5.
I. Pak, Partition identities and geometric bijections, Proc. Amer. Math. Soc. 132 (2004), 3457-3462.
Hao Pan and Zhi-Wei Sun, A combinatorial identity with application to Catalan numbers, arXiv:math/0509648 [math.CO], 2005-2006.
Murray Tannock, Equivalence classes of mesh patterns with a dominating pattern, MSc Thesis, Reykjavik Univ., May 2016. See Appendix B2.
Kevin Topley, Computationally Efficient Bounds for the Sum of Catalan Numbers, arXiv:1601.04223 [math.CO], 2016.

Cf. A000108, A000245, A000984, A001246, A002476, A002897, A006134, A033536, A045309, A079727, A082894, A094638, A094639, A128287, A358436.

[(&+[Catalan(k): k in [0..n]]): n in [0..40]]; // G. C. Greubel, Jun 30 2024

a:= proc(n) option remember; `if`(n<2, n+1,
      ((5*n-1)*a(n-1)-(4*n-2)*a(n-2))/(n+1))
    end:
seq(a(n), n=0..30);  # Alois P. Heinz, May 18 2013
A014137List := proc(m) local A, P, n; A := [1]; P := [1];
for n from 1 to m - 2 do P := ListTools:-PartialSums([op(P), P[-n]]);
A := [op(A), P[-1]] od; A end: A014137List(30); # Peter Luschny, Mar 26 2022

Table[Sum[(2k)!/(k!)^2/(k+1),{k,0,n}],{n,0,30}] (* Alexander Adamchuk, Jul 11 2006 *)
Accumulate[CatalanNumber[Range[0,30]]] (* Harvey P. Dale, May 08 2012 *)
a[ n_] := SeriesCoefficient[ (1 - (1 - 4 x)^(1/2)) / (2 x (1 - x)), {x, 0, n}]; (* Michael Somos, Oct 24 2015 *)
Table[(1 + CatalanNumber[n] (3 (n + 1) Hypergeometric2F1[1, -n, 1/2 - n, 1/4] - 4 n - 2))/2, {n, 0, 20}] (* Vladimir Reshetnikov, Oct 03 2016 *)

Vec((1-(1-4*x)^(1/2))/(2*x*(1-x))+O(x^99)) \\ Charles R Greathouse IV, Feb 11 2011

sm(v)={my(s=vector(#v)); s[1]=v[1]; for(n=2, #v, s[n]=v[n]+s[n-1]); s; }
C(n)=binomial(2*n, n)/(n+1);
sm(vector(66, n, C(n-1)))
/* Joerg Arndt, May 04 2013 */

from _future_ import division
A014137_list, b, s = [], 1, 0
for n in range(10**2):
    s += b
    A014137_list.append(s)
    b = b*(4*n+2)//(n+2) # Chai Wah Wu, Jan 28 2016

def A014137():
    f, c, n = 1, 1, 1
    while True:
        yield f
        n += 1
        c = c * (4*n - 6) // n
        f = c + f
a = A014137()
print([next(a) for  in range(29)]) # _Peter Luschny, Nov 30 2016

a(n) = A014138(n-1) + 1.
G.f.: (1 - (1 - 4*x)^(1/2))/(2*x*(1 - x)).
a(n) = Sum_{k=0..n} (2k)!/(k!)^2/(k+1). - Alexander Adamchuk, Jul 11 2006
D-finite with recurrence: (n+1)*a(n) + (1-5*n)*a(n-1) + 2*(2*n-1)*a(n-2) = 0. - R. J. Mathar, Dec 14 2011
Mathar's formula reduces to 2*(2*n-1)*C(n-1) = (n+1)*C(n), which is a known recurrence of the Catalan numbers, so the conjecture is true. - Peter J. Taylor, Mar 23 2015
Let C(n+1) = binomial(2*n+2,n+1)/(n+2) and H(n) = hypergeometric([1,n+3/2],[n+3],4) then A014137(n) = -(-1)^(2/3) - C(n+1)*H(n) and A014138(n) = -I^(2/3) - C(n+1)*H(n). - Peter Luschny, Aug 09 2012
G.f. (conjecture): Q(0)/(1-x), where Q(k)= 1 + (4*k + 1)*x/(k + 1 - 2*x*(k + 1)*(4*k + 3)/(2*x*(4*k + 3) + (2*k + 3)/Q(k+1))); (continued fraction). - Sergei N. Gladkovskii, May 14 2013
a(n) ~ 2^(2*n + 2)/(3*sqrt(Pi)*n^(3/2)). - Vaclav Kotesovec, Dec 10 2013
0 =  a(n)*(16*a(n+1) - 26*a(n+2) + 10*a(n+3)) + a(n+1)*(-14*a(n+1) + 23*a(n+2) - 11*a(n+3)) + a(n+2)*(a(n+2) + a(n+3)) if n >= 0. - Michael Somos, Oct 24 2015
a(n) = (1 + A000108(n)*(3*(n+1)*hypergeom([1,-n], [1/2-n], 1/4) - 4*n - 2))/2. - Vladimir Reshetnikov, Oct 03 2016
G.f. A(x) satisfies: A(x) = 1 / (1 - x) + x * (1 - x) * A(x)^2. - Ilya Gutkovskiy, Jul 25 2021
From Peter Luschny, Nov 16 2022: (Start)
a(n) = C(n)*hypergeom([1, -n - 1], [1/2 - n], 1/4) + 1/2.
a(n) = A358436(n) / C(n). (End)
E.g.f.: exp(2*x)*(BesselI(0, 2*x)/2 - BesselI(1, 2*x)) + exp(x)/2*(3*Integral_{x=-oo..oo} BesselI(0,2*x)*exp(x) dx + 1). - Mélika Tebni, Sep 01 2024

A094638 Triangle read by rows: T(n,k) = |s(n,n+1-k)|, where s(n,k) are the signed Stirling numbers of the first kind A008276 (1 <= k <= n; in other words, the unsigned Stirling numbers of the first kind in reverse order).

Original entry on oeis.org

1, 1, 1, 1, 3, 2, 1, 6, 11, 6, 1, 10, 35, 50, 24, 1, 15, 85, 225, 274, 120, 1, 21, 175, 735, 1624, 1764, 720, 1, 28, 322, 1960, 6769, 13132, 13068, 5040, 1, 36, 546, 4536, 22449, 67284, 118124, 109584, 40320, 1, 45, 870, 9450, 63273, 269325, 723680, 1172700, 1026576, 362880

Offset: 1

André F. Labossière, May 17 2004

Triangle of coefficients of the polynomial (x+1)(x+2)...(x+n), expanded in decreasing powers of x. - T. D. Noe, Feb 22 2008
Row n also gives the number of permutation of 1..n with complexity 0,1,...,n-1. See the comments in A008275. - N. J. A. Sloane, Feb 08 2019
T(n,k) is the number of deco polyominoes of height n and having k columns. A deco polyomino is a directed column-convex polyomino in which the height, measured along the diagonal, is attained only in the last column. Example: T(2,1)=1 and T(2,2)=1 because the deco polyominoes of height 2 are the vertical and horizontal dominoes, having, respectively, 1 and 2 columns. - Emeric Deutsch, Aug 14 2006
Sum_{k=1..n} k*T(n,k) = A121586. - Emeric Deutsch, Aug 14 2006
Let the triangle U(n,k), 0 <= k <= n, read by rows, be given by [1,0,1,0,1,0,1,0,1,0,1,...] DELTA [1,1,2,2,3,3,4,4,5,5,6,...] where DELTA is the operator defined in A084938; then T(n,k) = U(n-1,k-1). - Philippe Deléham, Jan 06 2007
From Tom Copeland, Dec 15 2007: (Start)
Consider c(t) = column vector(1, t, t^2, t^3, t^4, t^5, ...).
Starting at 1 and sampling every integer to the right, we obtain (1,2,3,4,5,...). And T * c(1) = (1, 1*2, 1*2*3, 1*2*3*4,...), giving n! for n > 0. Call this sequence the right factorial (n+)!.
Starting at 1 and sampling every integer to the left, we obtain (1,0,-1,-2,-3,-4,-5,...). And T * c(-1) = (1, 1*0, 1*0*-1, 1*0*-1*-2,...) = (1, 0, 0, 0, ...), the left factorial (n-)!.
Sampling every other integer to the right, we obtain (1,3,5,7,9,...). T * c(2) = (1, 1*3, 1*3*5, ...) = (1,3,15,105,945,...), giving A001147 for n > 0, the right double factorial, (n+)!!.
Sampling every other integer to the left, we obtain (1,-1,-3,-5,-7,...). T * c(-2) = (1, 1*-1, 1*-1*-3, 1*-1*-3*-5,...) = (1,-1,3,-15,105,-945,...) = signed A001147, the left double factorial, (n-)!!.
Sampling every 3 steps to the right, we obtain (1,4,7,10,...). T * c(3) = (1, 1*4, 1*4*7,...) = (1,4,28,280,...), giving A007559 for n > 0, the right triple factorial, (n+)!!!.
Sampling every 3 steps to the left, we obtain (1,-2,-5,-8,-11,...), giving T * c(-3) = (1, 1*-2, 1*-2*-5, 1*-2*-5*-8,...) = (1,-2,10,-80,880,...) = signed A008544, the left triple factorial, (n-)!!!.
The list partition transform A133314 of [1,T * c(t)] gives [1,T * c(-t)] with all odd terms negated; e.g., LPT[1,T*c(2)] = (1,-1,-1,-3,-15,-105,-945,...) = (1,-A001147). And e.g.f. for [1,T * c(t)] = (1-xt)^(-1/t).
The above results hold for t any real or complex number. (End)
Let R_n(x) be the real and I_n(x) the imaginary part of Product_{k=0..n} (x + I*k). Then, for n=1,2,..., we have R_n(x) = Sum_{k=0..floor((n+1)/2)}(-1)^k*Stirling1(n+1,n+1-2*k)*x^(n+1-2*k), I_n(x) = Sum_{k=0..floor(n/2)}(-1)^(k+1)*Stirling1(n+1,n-2*k)*x^(n-2*k). - Milan Janjic, May 11 2008
T(n,k) is also the number of permutations of n with "reflection length" k (i.e., obtained from 12..n by k not necessarily adjacent transpositions). For example, when n=3, 132, 213, 321 are obtained by one transposition, while 231 and 312 require two transpositions. - Kyle Petersen, Oct 15 2008
From Tom Copeland, Nov 02 2010: (Start)
[x^(y+1) D]^n = x^(n*y) [T(n,1)(xD)^n + T(n,2)y (xD)^(n-1) + ... + T(n,n)y^(n-1)(xD)], with D the derivative w.r.t. x.
E.g., [x^(y+1) D]^4 = x^(4*y) [(xD)^4 + 6 y(xD)^3 + 11 y^2(xD)^2 + 6 y^3(xD)].
(xD)^m can be further expanded in terms of the Stirling numbers of the second kind and operators of the form x^j D^j. (End)
With offset 0, 0 <= k <= n: T(n,k) is the sum of products of each size k subset of {1,2,...,n}. For example, T(3,2) = 11 because there are three subsets of size two: {1,2},{1,3},{2,3}. 1*2 + 1*3 + 2*3 = 11. - Geoffrey Critzer, Feb 04 2011
The Kn11, Fi1 and Fi2 triangle sums link this triangle with two sequences, see the crossrefs. For the definitions of these triangle sums see A180662. The mirror image of this triangle is A130534. - Johannes W. Meijer, Apr 20 2011
T(n+1,k+1) is the elementary symmetric function a_k(1,2,...,n), n >= 0, k >= 0, (a_0(0):=1). See the T. D. Noe and Geoffrey Critzer comments given above. For a proof see the Stanley reference, p. 19, Second Proof. - Wolfdieter Lang, Oct 24 2011
Let g(t) = 1/d(log(P(j+1,-t)))/dt (see Tom Copeland's 2007 formulas). The Mellin transform (t to s) of t*Dirac[g(t)] gives Sum_{n=1..j} n^(-s), which as j tends to infinity gives the Riemann zeta function for Re(s) > 1. Dirac(x) is the Dirac delta function. The complex contour integral along a circle of radius 1 centered at z=1 of z^s/g(z) gives the same result. - Tom Copeland, Dec 02 2011
Rows are coefficients of the polynomial expansions of the Pochhammer symbol, or rising factorial, Pch(n,x) = (x+n-1)!/(x-1)!. Expansion of Pch(n,xD) = Pch(n,Bell(.,:xD:)) in a polynomial with terms :xD:^k=x^k*D^k gives the Lah numbers A008297. Bell(n,x) are the unsigned Bell polynomials or Stirling polynomials of the second kind A008277. - Tom Copeland, Mar 01 2014
From Tom Copeland, Dec 09 2016: (Start)
The Betti numbers, or dimension, of the pure braid group cohomology. See pp. 12 and 13 of the Hyde and Lagarias link.
Row polynomials and their products appear in presentation of the Jack symmetric functions of R. Stanley. See Copeland link on the Witt differential generator.
(End)
From Tom Copeland, Dec 16 2019: (Start)
The e.g.f. given by Copeland in the formula section appears in a combinatorial Dyson-Schwinger equation of quantum field theory in Yeats in Thm. 2 on p. 62 related to a Hopf algebra of rooted trees. See also the Green function on p. 70.
Per comments above, this array contains the coefficients in the expansion in polynomials of the Euler, or state number, operator xD of the rising factorials Pch(n,xD) = (xD+n-1)!/(xD-1)! = x [:Dx:^n/n!]x^{-1} = L_n^{-1}(-:xD:), where :Dx:^n = D^n x^n and :xD:^n = x^n D^n. The polynomials L_n^{-1} are the Laguerre polynomials of order -1, i.e., normalized Lah polynomials.
The Witt differential operators L_n = x^(n+1) D and the row e.g.f.s appear in Hopf and dual Hopf algebra relations presented by Foissy. The Witt operators satisfy L_n L_k - L_k L_n = (k-n) L_(n+k), as for the dual Hopf algebra. (End)

			Triangle starts:
  1;
  1,  1;
  1,  3,  2;
  1,  6, 11,  6;
  1, 10, 35, 50, 24;
...

M. Miyata and J. W. Son, On the complexity of permutations and the metric space of bijections, Tensor, 60 (1998), No. 1, 109-116 (MR1768839).
Jerome Spanier and Keith B. Oldham, "Atlas of Functions", Hemisphere Publishing Corp., 1987, chapter 18, equations 18:4:2 - 18:4:8 at page 151.
R. P. Stanley, Enumerative Combinatorics, Vol. 1, Cambridge University Press, 1997.

T. D. Noe, Rows n = 1..51 of triangle, flattened
E. Barcucci, A. Del Lungo and R. Pinzani, "Deco" polyominoes, permutations and random generation, Theoretical Computer Science, 159, 1996, 29-42.
F. Bergeron, Ph. Flajolet and B. Salvy, Varieties of Increasing Trees, Lecture Notes in Computer Science vol. 581, ed. J.-C. Raoult, Springer 1992, pp. 24-48. Added Mar 01 2014
F. Bergeron, Philippe Flajolet and Bruno Salvy, Varieties of increasing trees, HAL, Rapport De Recherche Inria. Added Mar 01 2014
Tom Copeland, Addendum to Mathemagical Forests
Tom Copeland, A Class of Differential Operators and the Stirling Numbers
Tom Copeland, Mellin Interpolation of Differential Ops and Associated Infinigens and Appell Polynomials: The Ordered, Laguerre, and Scherk-Witt-Lie Diff Ops
Tom Copeland, Witt differential generator for special Jack symmetric functions / polynomials
FindStat - Combinatorial Statistic Finder, The absolute length of a permutation
L. Foissy, Faa di Bruno subalgebras of the Hopf algebra of planar trees from combinatorial Dyson-Schwinger equations, arXiv:0707.1204 [math.RA], (2007).
O. Furdui, T. Trif, On the Summation of Certain Iterated Series, J. Int. Seq. 14 (2011) #11.6.1
F. Hivert, J.-C. Novelli and J.-Y. Thibon, The Algebra of Binary Search Trees, Theoretical Computer Science, 339 (2005), 129-165.
T. Hyde and J. Lagarias Polynomial splitting measures and cohomology of the pure braid group, arXiv preprint arXiv:1604.05359 [math.RT], 2016.
MathOverflow, Motivation of Virasoro algebra, an answer by Tom Copeland to an MO question posed in 2012.
Romeo Mestrovic, Lucas' theorem: its generalizations, extensions and applications (1878--2014), arXiv preprint arXiv:1409.3820 [math.NT], 2014.
Robert E. Moritz, On the sum of products of n consecutive integers, Univ. Washington Publications in Math., 1 (No. 3, 1926), 44-49 [Annotated scanned copy]
M. D. Schmidt, Generalized j-Factorial Functions, Polynomials, and Applications , J. Int. Seq. 13 (2010), 10.6.7.
M. Z. Spivey, On Solutions to a General Combinatorial Recurrence, J. Int. Seq. 14 (2011) # 11.9.7.
K. Yeats, A Combinatorial Perspective on Quantum Field Theory, SpringerBriefs in Mathematical Physics, Vol. 15, 2017.

A008276 gives the (signed) Stirling numbers of the first kind.
Cf. A000108, A014137, A001246, A033536, A000984, A094639, A006134, A082894, A002897, A079727, A000217 (2nd column), A000914 (3rd column), A001303 (4th column), A000915 (5th column), A053567 (6th column), A000142 (row sums).
Triangle sums (see the comments): A124380 (Kn11), A001710 (Fi1, Fi2). - Johannes W. Meijer, Apr 20 2011
Cf. A121586, A130534, A143491.

Flat(List([1..10], n-> List([1..n], k-> Stirling1(n,n-k+1) ))); # G. C. Greubel, Dec 29 2019

a094638 n k = a094638_tabl !! (n-1) !! (k-1)
a094638_row n = a094638_tabl !! (n-1)
a094638_tabl = map reverse a130534_tabl
-- Reinhard Zumkeller, Aug 01 2014

[(-1)^(k+1)*StirlingFirst(n,n-k+1): k in [1..n], n in [1..10]]; // G. C. Greubel, Dec 29 2019

T:=(n,k)->abs(Stirling1(n,n+1-k)): for n from 1 to 10 do seq(T(n,k),k=1..n) od; # yields sequence in triangular form. # Emeric Deutsch, Aug 14 2006

Table[CoefficientList[Series[Product[1 + i x, {i,n}], {x,0,20}], x], {n,0,6}] (* Geoffrey Critzer, Feb 04 2011 *)
Table[Abs@StirlingS1[n, n-k+1], {n, 10}, {k, n}]//Flatten (* Michael De Vlieger, Aug 29 2015 *)

create_list(abs(stirling1(n+1,n-k+1)),n,0,10,k,0,n); /* Emanuele Munarini, Jun 01 2012 */

{T(n,k)=if(n<1 || k>n,0,(n-1)!*polcoeff(polcoeff(x*y/(1 - x*y+x*O(x^n))^(1 + 1/y),n,x),k,y))} /* Paul D. Hanna, Jul 21 2011 */

[[stirling_number1(n, n-k+1) for k in (1..n)] for n in (1..10)] # G. C. Greubel, Dec 29 2019

With P(n,t) = Sum_{k=0..n-1} T(n,k+1) * t^k = 1*(1+t)*(1+2t)...(1+(n-1)*t) and P(0,t)=1, exp[P(.,t)*x] = (1-tx)^(-1/t). T(n,k+1) = (1/k!) (D_t)^k (D_x)^n [ (1-tx)^(-1/t) - 1 ] evaluated at t=x=0. (1-tx)^(-1/t) - 1 is the e.g.f. for a plane m-ary tree when t=m-1. See Bergeron et al. in "Varieties of Increasing Trees". - Tom Copeland, Dec 09 2007
First comment and formula above rephrased as o.g.f. for row n: Product_{i=0...n} (1+i*x). - Geoffrey Critzer, Feb 04 2011
n-th row polynomials with alternate signs are the characteristic polynomials of the (n-1)x(n-1) matrices with 1's in the superdiagonal, (1,2,3,...) in the main diagonal, and the rest zeros. For example, the characteristic polynomial of [1,1,0; 0,2,1; 0,0,3] is x^3 - 6*x^2 + 11*x - 6. - Gary W. Adamson, Jun 28 2011
E.g.f.: A(x,y) = x*y/(1 - x*y)^(1 + 1/y) = Sum_{n>=1, k=1..n} T(n,k)*x^n*y^k/(n-1)!. - Paul D. Hanna, Jul 21 2011
With F(x,t) = (1-t*x)^(-1/t) - 1 an e.g.f. for the row polynomials P(n,t) of A094638 with P(0,t)=0, G(x,t)= [1-(1+x)^(-t)]/t is the comp. inverse in x. Consequently, with H(x,t) = 1/(dG(x,t)/dx) = (1+x)^(t+1),
  P(n,t) = [(H(x,t)*d/dx)^n] x evaluated at x=0; i.e.,
  F(x,t) = exp[x*P(.,t)] = exp[x*H(u,t)*d/du] u, evaluated at u = 0.
  Also, dF(x,t)/dx = H(F(x,t),t). - Tom Copeland, Sep 20 2011
T(n,k) = |A008276(n,k)|. - R. J. Mathar, May 19 2016
The row polynomials of this entry are the reversed row polynomials of A143491 multiplied by (1+x). E.g., (1+x)(1 + 5x + 6x^2) = (1 + 6x + 11x^2 + 6x^3). - Tom Copeland, Dec 11 2016
Regarding the row e.g.f.s in Copeland's 2007 formulas, e.g.f.s for A001710, A001715, and A001720 give the compositional inverses of the e.g.f. here for t = 2, 3, and 4 respectively. - Tom Copeland, Dec 28 2019

Edited by Emeric Deutsch, Aug 14 2006

A094639 Partial sums of squares of Catalan numbers (A000108).

Original entry on oeis.org

1, 2, 6, 31, 227, 1991, 19415, 203456, 2248356, 25887400, 307993016, 3763786812, 47032778956, 598933188956, 7751562502556, 101741582076581, 1351906409905481, 18159677984049581, 246298405721739581

Offset: 0

André F. Labossière, May 27 2004

Koshy and Salmassi give an elementary proof that the only prime Catalan numbers are A000108(2) = 2 and A000108(3) = 5. Franklin T. Adams-Watters showed that the only semiprime Catalan number is A000108(4) = 14. The subsequence of primes in the partial sum of squares of Catalan numbers begins: 2, 31, 227, 101741582076581. [Jonathan Vos Post, May 27 2010]

Conjecture: For any positive integer n, the polynomial P_n(x) = sum_{k = 0}^n(C_k)^2*x^k (with C_k = binomial(2k, k)/(k+1)) is irreducible over the field of rational numbers. [Zhi-Wei Sun, Mar 23 2013]

Michael De Vlieger, Table of n, a(n) for n = 0..838
Paul Barry, Jacobsthal Decompositions of Pascal's Triangle, Ternary Trees, and Alternating Sign Matrices, Journal of Integer Sequences, 19, 2016, #16.3.5.
Joel E. Cohen, Variance Functions of Asymptotically Exponentially Increasing Integer Sequences Go Beyond Taylor's Law, J. Int. Seq., Vol. 25 (2022), Article 22.9.3.

Cf. A000108, A094638, A014137, A001246, A033536, A000984, A006134, A082894, A002897, A079727.

Accumulate[CatalanNumber[Range[0,20]]^2] (* Harvey P. Dale, May 01 2011 *)

a(n) = Sum_{k=0..n} ((2k)!/(k!)^2/(k+1))^2. - Alexander Adamchuk, Feb 16 2008
Sum_{i=1..n} [c(i)]^2 = Sum_{i=1..n} [C(2*i-2, i-1)/i]^2 = (1/(n-1)!)^2 * [ n^C(2*n-4, 1) + {2*C(n-1, 2)}*n^(2*n-5) + {C(n-2, 0) + 4*C(n-2, 1) + 13*C(n-2, 2) + 22*C(n-2, 3) + 12*C(n-2, 4)}*n^C(2*n-6, 1) + {12*C(n-3, 1) + 152*C(n-3, 2) + 458*C(n-3, 3) + 640*C(n-3, 4) + 440*C(n-3, 5) + 120*C(n-3, 6)}*n^(2*n-7) + {40*C(n-4, 0) + 313*C(n-4, 1) + 2332*C(n-4, 2) + 9536*C(n-4, 3) + 21409*C(n-4, 4) + 28068*C(n-4, 5) + 21700*C(n-4, 6) + 9240*C(n-4, 7) + 1680*C(n-4, 8) + ... + C(n-3, 0)*((n-1)!)^2 ].
Recurrence: (n+1)^2*a(n) = (17*n^2 - 14*n + 5)*a(n-1) - 4*(2*n - 1)^2*a(n-2). - Vaclav Kotesovec, Jul 01 2016
a(n) ~ 2^(4*n+4) /(15*Pi*n^3). - Vaclav Kotesovec, Jul 01 2016

A244500 Number T(n, k) of ways to place k points on an n X n X n triangular grid so that no pair of them has distance sqrt(3). Triangle read by rows.

Original entry on oeis.org

1, 1, 1, 3, 3, 1, 1, 6, 12, 8, 1, 10, 36, 55, 33, 9, 1, 15, 87, 248, 378, 339, 187, 63, 12, 1, 1, 21, 180, 820, 2190, 3606, 3716, 2340, 825, 125, 1, 28, 333, 2212, 9110, 24474, 43928, 53018, 42774, 22792, 7945, 1764, 196, 1, 36, 567, 5163, 30300, 121077, 339621

Offset: 1

Heinrich Ludwig, Jun 29 2014

In the following triangular grid points x have Euclidean distance sqrt(3) from point o. It is the second closest distance possible among grid points.
      x
     . .
    . o .
   x . . x
Triangle T(n, k) is irregular: 0 <= k <= max(n), where max(n), the maximal number of points that can be placed on the grid, is:
  for n = 3j-2:            max(n) = A000326(j) = j(3j-1)/2;
  for n = 3j-1 or n = 3j:  max(n) = A045943(j) = 3j(j+1)/2; j = 1,2,3,...
Empirical: (1) The number of ways to place the maximal number of points for grid sizes n = 3j are cubes of Catalan numbers, i.e., for n = 3j: T(n, max(n)) = C(j+1)^3 = A033536(j+1). (2) For n = 3j-2: T(n, max(n)) = A244506(n) = A244507^2(n). (3) For n = 3j-1: T(n, max(n)) = A000012(n) = 1 and T(n, max(n)-1) = 3j^2.
Row n is also the coefficients of the independence polynomial of the n-triangular honeycomb acute knight graph. - Eric W. Weisstein, May 21 2017

			On an 8 X 8 X 8 grid there is T(8, 18) = 1 way to place 18 points (x) so that no pair of points has the distance square root of 3.
         x
        x x
       . . .
      x . . x
     x x . x x
    . . . . . .
   x . . x . . x
  x x . x x . x x
Continuation of this pattern will give the unique maximal solution for all n = 3j-1.
Triangle T(n, k) begins:
  1,  1;
  1,  3,   3,   1;
  1,  6,  12,   8;
  1, 10,  36,  55,   33,    9;
  1, 15,  87, 248,  378,  339,  187,   63,  12,   1;
  1, 21, 180, 820, 2190, 3606, 3716, 2340, 825, 125;
First row refers to n = 1.

Heinrich Ludwig, Table of n, a(n) for n = 1..153
Stan Wagon, Graph Theory Problems from Hexagonal and Traditional Chess, The College Mathematics Journal, Vol. 45, No. 4, September 2014, pp. 278-287.
Eric Weisstein's World of Mathematics, Independence Polynomial

Cf. A000108, A000326, A033536,  A045943, A244506, A244507.
Cf. A000217 (column 2), A086274 (1/3 * column 3), A244501 (column 4), A244502 (column 5), A244503 (column 6).
Cf. A287195 (length of row n). - Eric W. Weisstein, May 21 2017
Cf. A287204 (row sums). - Eric W. Weisstein, May 21 2017

A167892 a(n) = Sum_{k=1..n} Catalan(k)^2.

Original entry on oeis.org

1, 5, 30, 226, 1990, 19414, 203455, 2248355, 25887399, 307993015, 3763786811, 47032778955, 598933188955, 7751562502555, 101741582076580, 1351906409905480, 18159677984049580, 246298405721739580, 3369517588450715680, 46457194476711692080

Offset: 1

Alexander Adamchuk, Nov 15 2009

nonn

CatalanNumber(k) = (2k)!/(k!*(k+1)!) = binomial(2k,k)/(k+1).

G. C. Greubel, Table of n, a(n) for n = 1..500
Paul Barry, Jacobsthal Decompositions of Pascal's Triangle, Ternary Trees, and Alternating Sign Matrices, Journal of Integer Sequences, 19, 2016, #16.3.5.
Eric Weisstein's World of Mathematics, Catalan Number

Cf. A000108, A014138, A167892, A167893, A001246, A033536, A014137, A094639.

[&+[Catalan(i)^2: i in [1..n]]: n in [1..20]]; // Vincenzo Librandi, Jul 01 2016

Array[n \[Function] Sum[CatalanNumber[k]^2, {k, 1, n}], 20] (* J. Mulder (jasper.mulder(AT)planet.nl), Jan 25 2010 *)
Accumulate[CatalanNumber[Range[1, 20]]^2] (* Vincenzo Librandi, Jul 01 2016 *)

a(n) = Sum_{k=1..n} Catalan(k)^2.
a(n) = Sum_{k=1..n} ((2k)!/(k!*(k+1)!))^2.
a(n) = Sum_{k=1..n} A000108(k)^2.
a(n) = Sum_{k=1..n} A001246(k).
a(n) = A094639(n) - 1.
G.f.: (Hypergeometric2F1(-1/2,-1/2,1,16*x) - 4*x - 1)/(4*x*(1 - x)). - Ilya Gutkovskiy, Jul 01 2016

More terms from J. Mulder (jasper.mulder(AT)planet.nl), Jan 25 2010

A167893 a(n) = Sum_{k=1..n} Catalan(k)^3.

Original entry on oeis.org

1, 9, 134, 2878, 76966, 2376934, 81330523, 3005537523, 117938569451, 4856184495787, 208008478587443, 9208478072445171, 419215292661445171, 19548493234125829171, 930767164551264230296, 45133682592532326893296, 2224173698690413601132296, 111192059034974606204132296

Offset: 1

Alexander Adamchuk, Nov 15 2009

nonn

Catalan(k) = A000108(k) = (2k)!/(k!*(k+1)!) = C(2*k,k)/(k+1).
For prime p=7, p^2 divides a(p^2), and p divides all a(n) for n from (p^2-1)/2 to p^2-2.
For prime p=19 or 97, p divides all a(n) for n from (p-1)/2 to p-2.

G. C. Greubel, Table of n, a(n) for n = 1..500
Eric Weisstein's World of Mathematics, Catalan Number

Cf. A000108, A014138, A167892, A167893, A001246, A033536, A014137, A094639.

[&+[Catalan(i)^3: i in [1..n]]: n in [1..20]]; // Vincenzo Librandi, Jul 01 2016

Array[n \[Function] Sum[CatalanNumber[k]^3, {k, 1, n}], 15] (* J. Mulder (jasper.mulder(AT)planet.nl), Jan 25 2010 *)
Accumulate[CatalanNumber[Range[1, 20]]^3] (* Vincenzo Librandi, Jul 01 2016 *)

a(n)=sum(k=1,n,(binomial(k+k,k)/(k+1))^3) /* Charles R Greathouse IV, Jun 14 2011 */

a(n) = Sum_{k=1..n} A033536(k).
Recurrence: (n+1)^3*a(n) = (5*n - 1)*(13*n^2 - 16*n + 7)*a(n-1) - 8*(2*n - 1)^3*a(n-2). - Vaclav Kotesovec, Jul 01 2016
a(n) ~ 2^(6*n+6) / (63*Pi^(3/2)*n^(9/2)). - Vaclav Kotesovec, Jul 01 2016

More terms from J. Mulder, (jasper.mulder(AT)planet.nl), Jan 25 2010

More terms from Sean A. Irvine, Jun 13 2011

A244506 Number of ways to place the maximal number of points that can be placed on a j X j X j triangular grid, j=3n-2, so that no pair of them has distance sqrt(3).

Original entry on oeis.org

1, 9, 196, 6084, 219024, 8450649, 338265664, 13840346025, 574510941225, 24093764931600

Offset: 1

Heinrich Ludwig, Jul 11 2014

(1) All a(n) are square numbers. The sequence of their roots is A244507.
(2) On a j X j X j grid, j = 3n-2, the maximal number of points that can be placed is the pentagonal number A000326(n).
(3) On a maximally occupied grid, the following grid points "X" are always occupied (example for j = 7, for other j's expand this pattern):
              X
             . .
            . . .
           x . . X
          . . . . .
         . . . . . .
        X . . X . . X
(4) For j X j X j grids, j = 3n, the corresponding numbers are cubes of Catalan numbers, A033536(n). For j = 3n-1, the corresponding numbers are always 1, A000012(n).

Eric Weisstein's World of Mathematics, Maximum Independent Vertex Set

Cf. A000326, A244507, A244500.

Cf. A297537 (maximum independent vertex sets for n-triangular honeycomb acute knight graph).

a(10) from Heinrich Ludwig, Aug 17 2014

A297537 Number of maximum independent vertex sets in the n-triangular honeycomb grid graph.

Original entry on oeis.org

1, 1, 8, 9, 1, 125, 196, 1, 2744, 6084, 1, 74088, 219024, 1, 2299968, 8450649, 1, 78953589, 338265664, 1, 2924207000, 13840346025, 1, 114933031928, 574510941225, 1, 4738245926336, 24093764931600, 1, 203152294091656

Offset: 1

Eric W. Weisstein, Dec 31 2017

nonn

Eric Weisstein's World of Mathematics, Maximum Independent Vertex Set

Cf. A033536 (cubes of Catalan numbers).

Cf. A244506 (maximum independent vertex sets for mod(n,3) = 1).

a(3n) = A033536(n+1) = CatalanNumber(n+1)^3
a(3n-1) = 1
a(3n-2) = A244506(n).

cp's OEIS Frontend

A014137 Partial sums of Catalan numbers (A000108).

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A094638 Triangle read by rows: T(n,k) = |s(n,n+1-k)|, where s(n,k) are the signed Stirling numbers of the first kind A008276 (1 <= k <= n; in other words, the unsigned Stirling numbers of the first kind in reverse order).

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A094639 Partial sums of squares of Catalan numbers (A000108).

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A244500 Number T(n, k) of ways to place k points on an n X n X n triangular grid so that no pair of them has distance sqrt(3). Triangle read by rows.

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A167892 a(n) = Sum_{k=1..n} Catalan(k)^2.

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A167893 a(n) = Sum_{k=1..n} Catalan(k)^3.

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