A127158 -id:A127158 - cp's OEIS frontend

A026012 Second differences of Catalan numbers A000108.

1, 2, 6, 19, 62, 207, 704, 2431, 8502, 30056, 107236, 385662, 1396652, 5088865, 18642420, 68624295, 253706790, 941630580, 3507232740, 13105289370, 49114150020, 184560753390, 695267483664, 2625197720454, 9933364416572, 37660791173152, 143048202990504

Offset: 0

N. J. A. Sloane, Clark Kimberling

Number of (s(0), s(1), ..., s(n)) such that s(i) is a nonnegative integer and |s(i) - s(i-1)| = 1 for i = 1,2,...,n, s(0) = s(2n) = 2.
Number of Dyck paths of semilength n+2 with no initial and no final UD's. Example: a(2)=6 because the only Dyck paths of semilength 4 with no initial and no final UD's are UUDUDUDD, UUDUUDDD, UUUDDUDD, UUUDUDDD, UUDDUUDD, UUUUDDDD. - Emeric Deutsch, Oct 26 2003
Number of branches of length 1 starting from the root in all ordered trees with n+1 edges. Example: a(1)=2 because the tree /\ has two branches of length 1 starting from the root and the path-tree of length 2 has none. a(n) = Sum_{k=0..n+1} (k*A127158(n+1,k)). - Emeric Deutsch, Mar 01 2007
Number of staircase walks from (0,0) to (n,n) that never cross y=x+2. Example: a(3) = 19 because up,up,up,right,right,right is not allowed but the other binomial(6,3)-1 = 19 paths are. - Mark Spindler, Nov 11 2012
Number of standard Young tableaux of skew shape (n+2,n)/(2), for n>=2. - Ran Pan, Apr 07 2015

S. J. Cyvin and I. Gutman, Kekulé structures in benzenoid hydrocarbons, Lecture Notes in Chemistry, No. 46, Springer, New York, 1988 (see pp. 188, 196).

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Paul Barry, On a Central Transform of Integer Sequences, arXiv:2004.04577 [math.CO], 2020.
Reinis Cirpons, James East, and James D. Mitchell, Transformation representations of diagram monoids, arXiv:2411.14693 [math.RA], 2024. See pp. 3, 33.
Ryota Inagaki, Tanya Khovanova, and Austin Luo, On Chip-Firing on Undirected Binary Trees, Ann. Comb. (2025). See p. 15.
Jocelyn Quaintance and Harris Kwong, A combinatorial interpretation of the Catalan and Bell number difference tables, Integers, 13 (2013), #A29.
Murray Tannock, Equivalence classes of mesh patterns with a dominating pattern, MSc Thesis, Reykjavik Univ., May 2016. See Appendix B2.

T(2n, n), where T is the array defined in A026009.

Cf. A000108, A000245, A000344, A039599, A059346, A127158, A342912.

Differences[Table[CatalanNumber[n], {n, 0, 28}], 2] (* Jean-François Alcover, Sep 28 2012 *)
Table[Binomial[2n,n]-Binomial[2n,n-3],{n,0,26}] (* Mark Spindler, Nov 11 2012 *)

a(n) = 3*(3*n^2+3*n+2)*binomial(2*n, n)/((n+1)*(n+2)*(n+3)); /* Joerg Arndt, Aug 19 2012 */

Expansion of (1+x^1*C^3)*C^1, where C = (1-(1-4*x)^(1/2))/(2*x) is g.f. for Catalan numbers, A000108.
a(n) = 3*(3*n^2+3*n+2)*binomial(2*n, n)/((n+1)*(n+2)*(n+3)). - Emeric Deutsch, Oct 26 2003
a(n) = Sum_{k=0..2} A039599(n,k) = A000108(n) + A000245(n) + A000344(n). - Philippe Deléham, Nov 12 2008
a(n) = binomial(2*n,n)/(n+1)*hypergeom([-2,n+1/2],[n+2],4). - Peter Luschny, Aug 15 2012
a(n) = binomial(2*n,n) - binomial(2n,n-3). - Mark Spindler, Nov 11 2012
D-finite with recurrence (n+3)*a(n) + (-5*n-6)*a(n-1) + 2*(2*n-3)*a(n-2) = 0. - R. J. Mathar, Jun 20 2013
E.g.f.: exp(2*x)*(BesselI(0,2*x) - BesselI(3,2*x)). - Ilya Gutkovskiy, Feb 28 2017
Sum_{n>=0} a(n)/4^n = 6. - Amiram Eldar, Jul 10 2023
a(n) = C(n+2)+C(n)-2*C(n+1), C = A000108. - Alois P. Heinz, Apr 02 2025
Binomial transform of A342912. - Mélika Tebni, Apr 05 2025

A030238 Backwards shallow diagonal sums of Catalan triangle A009766.

Original entry on oeis.org

1, 1, 3, 7, 20, 59, 184, 593, 1964, 6642, 22845, 79667, 281037, 1001092, 3595865, 13009673, 47366251, 173415176, 638044203, 2357941142, 8748646386, 32576869203, 121701491701, 456012458965, 1713339737086, 6453584646837, 24364925260024, 92185136438942

Offset: 0

Wouter Meeussen

nonn

Number of linear forests of planted planar trees with n nodes (Christian G. Bower).
Number of ordered trees with n+2 edges and having no branches of length 1 starting from the root. Example: a(1)=1 because the only ordered tree with 3 edges having no branch of length 1 starting from the root is the path tree of length 3. a(n) = A127158(n+2,0). - Emeric Deutsch, Mar 01 2007
Hankel transform is A056520. - Paul Barry, Oct 16 2007

Vincenzo Librandi, Table of n, a(n) for n = 0..200
S. B. Ekhad and M. Yang, Proofs of Linear Recurrences of Coefficients of Certain Algebraic Formal Power Series Conjectured in the On-Line Encyclopedia Of Integer Sequences, (2017)
Taras Goy and Mark Shattuck, Hessenberg-Toeplitz Matrix Determinants with Schröder and Fine Number Entries, Carpathian Math. Publ., Vol. 15 (2023), No. 2, 420-436. See Theorem 3.

Cf. A000108, A009766, A127158, A132364.

g:=(1-sqrt(1-4*z))/z/(2-z+z*sqrt(1-4*z)): gser:=series(g,z=0,30): seq(coeff(gser,z,n),n=0..25); # Emeric Deutsch, Mar 01 2007

Sum[ triangle[ n-k, (n-k)-(k-1) ], {k, 1, Floor[ (n+1)/2 ]} ]
CoefficientList[Series[(1-Sqrt[1-4*x])/x/(2-x+x*Sqrt[1-4*x]), {x, 0, 20}], x] (* Vaclav Kotesovec, Feb 13 2014 *)

INVERT transform of 1, 2, 2, 5, 14, 42, 132, ... (cf. A000108).
a(n) = Sum_{k=0..floor(n/2)} (k+1)*binomial(2*n-3*k+1, n-k+1)/(2*n-3*k+1). Diagonal sums of A033184. - Paul Barry, Jun 22 2004
a(n) = Sum_{k=0..floor(n/2)} (k+1)*binomial(2*n-3*k, n-k)/(n-k+1). - Paul Barry, Feb 02 2005
G.f.: (1-sqrt(1-4*z))/(z*(2-z+z*sqrt(1-4*z))). - Emeric Deutsch, Mar 01 2007
G.f.: c(z)/(1-z^2*c(z)) where c(z) = (1-sqrt(1-4*z))/(2*z). - Ira M. Gessel, Sep 21 2020
D-finite with recurrence: (n+1)*a(n) + (-5*n+1)*a(n-1) + 2*(2*n-1)*a(n-2) + (n+1)*a(n-3) + 2*(-2*n+1)*a(n-4) = 0. - R. J. Mathar, Nov 30 2012
a(n) = Sum_{k=0..n} A000108(k)*A132364(n-k). - Philippe Deléham, Feb 27 2013
a(n) ~ 2^(2*n+6) / (49 * sqrt(Pi) * n^(3/2)). - Vaclav Kotesovec, Feb 13 2014

More terms from Christian G. Bower, Apr 15 1998

cp's OEIS Frontend

A026012 Second differences of Catalan numbers A000108.

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