A001791 -id:A001791 - cp's OEIS frontend

A000917 a(n) = (2n+3)!/(n!*(n+2)!).

3, 20, 105, 504, 2310, 10296, 45045, 194480, 831402, 3527160, 14872858, 62403600, 260757900, 1085822640, 4508102925, 18668849760, 77138650050, 318107374200, 1309542023790, 5382578744400, 22093039119060, 90567738003600, 370847442355650, 1516927277253024

Offset: 0

N. J. A. Sloane

G.f.: c(x)*(4-c(x))/(1-4*x)^(3/2), c(x) = g.f. for Catalan numbers A000108 (agrees with Hansen, 1975, p. 99, (5.27.9)). Convolution of A038679 with A000984 (central binomial coefficients); also convolution of A038665 with A000302 (powers of 4). - Wolfdieter Lang, Dec 11 1999
Appears as diagonal in A003506. - Zerinvary Lajos, Apr 12 2006
a(n) is the number of double rises in all Grand Dyck paths of semilength n+2. Example: a(0)=3 because in the 6 (=A000984(2)) Grand Dyck paths of semilength 2, namely udud, (uu)dd, uddu, d(uu)d, dudu, dd(uu), we have a total of 3 uu's (shown between parentheses). - Emeric Deutsch, Nov 29 2008

Eldon R. Hansen, A Table of Series and Products, Prentice-Hall, Englewood Cliffs, NJ, 1975, p. 99, (5.27.9).

T. D. Noe, Table of n, a(n) for n = 0..200
Jian Zhou, On Some Mathematics Related to the Interpolating Statistics, arXiv:2108.10514 [math-ph], 2021.

Cf. A001622, A001791, A007054, A038665, A038679, A000108, A000984, A000302, A003506, A073010.

1/beta(n, n+2) in A061928.

[(n+1)*Binomial(2*n+3, n+1): n in [0..25]]; // Vincenzo Librandi, Jun 01 2016

a := proc(n) (n+1)*binomial(2*n+3, n+2) end: seq(a(n), n=0..23); # Zerinvary Lajos, Nov 26 2006
seq((n+1)*binomial(2*n+4, n+2)/2, n=0..23); # Zerinvary Lajos, Feb 28 2007

Table[(2*n + 3)!/(n!*(n + 2)!), {n, 0, 25}] (* T. D. Noe, Jun 20 2012 *)

a(n) = (n+1)*binomial(2*n+3, n+1) = (n+1)*A001700(n+1). - Vincenzo Librandi, Jun 01 2016
a(n) = (2*n+3)*A001791(n+1). - R. J. Mathar, Nov 09 2021
D-finite with recurrence +(n+2)*a(n) +10*(-n-1)*a(n-1) +12*(2*n+1)*a(n-2)=0. - R. J. Mathar, Nov 09 2021
D-finite with recurrence n*(n+2)*a(n) -2*(2*n+3)*(n+1)*a(n-1)=0. - R. J. Mathar, Nov 09 2021
From Amiram Eldar, Jan 24 2022: (Start)
Sum_{n>=0} 1/a(n) = 1 - Pi/(3*sqrt(3)) = 1 - A073010.
Sum_{n>=0} (-1)^n/a(n) = 6*log(phi)/sqrt(5) - 1, where phi is the golden ratio (A001622). (End)

A025566 a(n) = number of (s(0), s(1), ..., s(n)) such that every s(i) is a nonnegative integer, s(0) = 0, s(1) = 1, |s(i) - s(i-1)| <= 1 for i >= 2. Also a(n) = sum of numbers in row n+1 of the array T defined in A026105. Also a(n) = T(n,n), where T is the array defined in A025564.

Original entry on oeis.org

1, 1, 1, 3, 8, 22, 61, 171, 483, 1373, 3923, 11257, 32418, 93644, 271219, 787333, 2290200, 6673662, 19478091, 56930961, 166613280, 488176938, 1431878079, 4203938697, 12353600427, 36331804089, 106932444885, 314946659951, 928213563878

Offset: 0

Clark Kimberling

nonn

a(n+1) is the number of Motzkin (2n)-paths whose last weak valley occurs immediately after step n. A weak valley in a Motzkin path (A001006) is an interior vertex whose following step has nonnegative slope and whose preceding step has nonpositive slope. For example, the weak valleys in the Motzkin path F.UF.FD.UD occur after the first, third and fifth steps as indicated by the dots (U=upstep of slope 1, D=downstep of slope -1, F=flatstep of slope 0) and, with n=2, a(3)=3 counts FFUD, UDUD, UFFD. - David Callan, Jun 07 2006

Starting with offset 2: (1, 3, 8, 22, 61, 171, 483, ...), = row sums of triangle A136537. - Gary W. Adamson, Jan 04 2008

Michael De Vlieger, Table of n, a(n) for n = 0..2100
Jean-Luc Baril, Richard Genestier, Sergey Kirgizov, Pattern distributions in Dyck paths with a first return decomposition constrained by height, arXiv:1911.03119 [math.CO], 2019.
C. Dalfó, M. A. Fiol, and N. López, New results for the Mondrian art problem, arXiv:2007.09639 [math.CO], 2020.
D. E. Davenport, L. W. Shapiro and L. C. Woodson, The Double Riordan Group, The Electronic Journal of Combinatorics, 18(2) (2012), #P33. - From _N. J. A. Sloane_, May 11 2012
Christian Krattenthaler, Daniel Yaqubi, Some determinants of path generating functions, II, arXiv:1802.05990 [math.CO], 2018; Adv. Appl. Math. 101 (2018), 232-265.
Donatella Merlini, Massimo Nocentini, Algebraic Generating Functions for Languages Avoiding Riordan Patterns, Journal of Integer Sequences, Vol. 21 (2018), Article 18.1.3.

First differences of A026135. Row sums of triangle A026105.
Pairwise sums of A005727. Column k=2 in A115990.
Cf. A001791, A132816.
Cf. A136537.

List([0..30],i->Sum([0..Int(i/2)],k->Binomial(i-2,k)*Binomial(i-k,k))); # Muniru A Asiru, Mar 09 2019

seq( sum('binomial(i-2,k)*binomial(i-k,k)', 'k'=0..floor(i/2)), i=0..30 ); # Detlef Pauly (dettodet(AT)yahoo.de), Nov 09 2001

CoefficientList[Series[x+(2x(x-1))/(1-3x-Sqrt[1-2x-3x^2]),{x,0,30}],x] (* Harvey P. Dale, Jun 12 2016 *)

G.f.: x + 2*x*(x-1)/(1-3x-sqrt(1-2x-3x^2)); for n > 1, first differences of the "directed animals" sequence A005773: a(n) = A005773(n) - A005773(n-1). - Emeric Deutsch, Aug 16 2002
Starting (1, 3, 8, 22, 61, 171, ...) gives the inverse binomial transform of A001791 starting (1, 4, 15, 56, 210, 792, ...). - Gary W. Adamson, Sep 01 2007
a(n) is the sum of the (n-2)-th row of triangle A131816. - Gary W. Adamson, Sep 01 2007
D-finite with recurrence n*a(n) +(-3*n+2)*a(n-1) +(-n+2)*a(n-2) +3*(n-4)*a(n-3)=0. - R. J. Mathar, Sep 15 2020

A029760 A sum with next-to-central binomial coefficients of even order, Catalan related.

Original entry on oeis.org

1, 8, 47, 244, 1186, 5536, 25147, 112028, 491870, 2135440, 9188406, 39249768, 166656772, 704069248, 2961699667, 12412521388, 51854046982, 216013684528, 897632738722, 3721813363288, 15401045060572, 63616796642368, 262357557683422, 1080387930269464

Offset: 0

Wolfdieter Lang

nonn

Proof by induction.
a(n) = total area below paths consisting of steps east (1,0) and north (0,1) from (0,0) to (n+2,n+2) that stay weakly below y=x. For example, the two paths with n=0 are
. _|.....|
|....._|
The first has area 1 below it, the second area 0 and so a(0)=1. - David Callan, Dec 09 2004
Convolution of A000346 with A001700. - Philippe Deléham, May 19 2009

Michael De Vlieger, Table of n, a(n) for n = 0..1657
Ran Pan, Jeffrey B. Remmel, Paired patterns in lattice paths, arXiv:1601.07988 [math.CO], 2016.
Sittipong Thamrongpairoj, Dowling Set Partitions, and Positional Marked Patterns, Ph. D. Dissertation, University of California-San Diego (2019).

Cf. A000108, A002457, A008549, A000984, A139262.

a[n_] := (n+3)^2 CatalanNumber[n+2]/2 - 2^(2n+3);
Table[a[n], {n, 0, 23}] (* Jean-François Alcover, Sep 25 2018 *)

a(n) = 4^(n+1)*Sum_{k=1..n+1} binomial(2k, k-1)/4^k = ((n+3)^2)*C(n+2)/2-2^(2*n+3), C = Catalan. Also a(n+1)=4*a(n)+binomial(2(n+2), n+1).
G.f.: (d/dx)c(x)/(1-4*x), where c(x) = g.f. for Catalan numbers; convolution of A001791 and powers of 4. G.f. also c(x)^2/(1-4*x)^(3/2); convolution of Catalan numbers A000108 C(n), n >= 1, with A002457; convolution of A008549(n), n >= 1, with A000984 (central binomial coefficients).
a(n) = Sum_{k=0..n+1} A039598(n+1,k)*k^2. - Philippe Deléham, Dec 16 2007

A100824 Number of partitions of n with at most one odd part.

Original entry on oeis.org

1, 1, 1, 2, 2, 4, 3, 7, 5, 12, 7, 19, 11, 30, 15, 45, 22, 67, 30, 97, 42, 139, 56, 195, 77, 272, 101, 373, 135, 508, 176, 684, 231, 915, 297, 1212, 385, 1597, 490, 2087, 627, 2714, 792, 3506, 1002, 4508, 1255, 5763, 1575, 7338, 1958, 9296, 2436, 11732, 3010, 14742

Offset: 0

Vladeta Jovovic, Jan 13 2005

From Gus Wiseman, Jan 21 2022: (Start)
Also the number of integer partitions of n with alternating sum <= 1, where the alternating sum of a sequence (y_1,...,y_k) is Sum_i (-1)^(i-1) y_i. These are the conjugates of partitions with at most one odd part. For example, the a(1) = 1 through a(9) = 12 partitions with alternating sum <= 1 are:
  1  11  21   22    32     33      43       44        54
         111  1111  221    2211    331      2222      441
                    2111   111111  2221     3311      3222
                    11111          3211     221111    3321
                                   22111    11111111  4311
                                   211111             22221
                                   1111111            33111
                                                      222111
                                                      321111
                                                      2211111
                                                      21111111
                                                      111111111
(End)

			From _Gus Wiseman_, Jan 21 2022: (Start)
The a(1) = 1 through a(9) = 12 partitions with at most one odd part:
  (1)  (2)  (3)   (4)   (5)    (6)    (7)     (8)     (9)
            (21)  (22)  (32)   (42)   (43)    (44)    (54)
                        (41)   (222)  (52)    (62)    (63)
                        (221)         (61)    (422)   (72)
                                      (322)   (2222)  (81)
                                      (421)           (432)
                                      (2221)          (441)
                                                      (522)
                                                      (621)
                                                      (3222)
                                                      (4221)
                                                      (22221)
(End)

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

Cf. A008951, A000070, A000097, A000098, A000710.
The case of alternating sum 0 (equality) is A000070.
A multiplicative version is A339846.
These partitions are ranked by A349150, conjugate A349151.
A000041 = integer partitions, strict A000009.
A027187 = partitions of even length, strict A067661, ranked by A028260.
A027193 = partitions of odd length, ranked by A026424.
A058695 = partitions of odd numbers.
A103919 = partitions by sum and alternating sum (reverse: A344612).
A277103 = partitions with the same number of odd parts as their conjugate.
Cf. A000984, A001791, A008549, A097805, A119620, A182616, A236559, A236913, A236914, A304620, A344607, A345958, A347443.

seq(coeff(convert(series((1+x/(1-x^2))/mul(1-x^(2*i),i=1..100),x,100),polynom),x,n),n=0..60); (C. Ronaldo)

nmax = 50; CoefficientList[Series[(1+x/(1-x^2)) * Product[1/(1-x^(2*k)), {k, 1, nmax}], {x, 0, nmax}], x] (* Vaclav Kotesovec, Mar 07 2016 *)
Table[Length[Select[IntegerPartitions[n],Count[#,?OddQ]<=1&]],{n,0,30}] (* _Gus Wiseman, Jan 21 2022 *)

a(n) = if(n%2==0, numbpart(n/2), sum(i=1, (n+1)\2, numbpart((n-2*i+1)\2))) \\ David A. Corneth, Jan 23 2022

G.f.: (1+x/(1-x^2))/Product(1-x^(2*i), i=1..infinity). More generally, g.f. for number of partitions of n with at most k odd parts is (1+Sum(x^i/Product(1-x^(2*j), j=1..i), i=1..k))/Product(1-x^(2*i), i=1..infinity).
a(n) ~ exp(sqrt(n/3)*Pi) / (2*sqrt(3)*n) if n is even and a(n) ~ exp(sqrt(n/3)*Pi) / (2*Pi*sqrt(n)) if n is odd. - Vaclav Kotesovec, Mar 07 2016
a(2*n) = A000041(n). a(2*n + 1) = A000070(n). - David A. Corneth, Jan 23 2022

More terms from C. Ronaldo (aga_new_ac(AT)hotmail.com), Jan 19 2005

A213343 1-quantum transitions in systems of N spin 1/2 particles, in columns by combination indices. Triangle read by rows, T(n, k) for n >= 1 and 0 <= k <= floor((n-1)/2).

Original entry on oeis.org

1, 4, 12, 3, 32, 24, 80, 120, 10, 192, 480, 120, 448, 1680, 840, 35, 1024, 5376, 4480, 560, 2304, 16128, 20160, 5040, 126, 5120, 46080, 80640, 33600, 2520, 11264, 126720, 295680, 184800, 27720, 462, 24576, 337920, 1013760, 887040, 221760, 11088

Offset: 1

Stanislav Sykora, Jun 09 2012

[General discussion]: Consider the 2^N numbers with N-digit binary expansion. Let a pair (v,w), here called a "transition", be such that there are exactly k+q digits which are '0' in v and '1' in w, and exactly k digits which are '1' in v and '0' in w. Then T(q;N,k) is the number of all such pairs.
For given N and q, the rows of the triangle T(q;N,k) sum up to Sum[k]T(q;N,k) = C(2N,N-q) which is the total number of q-quantum transitions or, equivalently, the number of pairs in which the sum of binary digits of w exceeds that of v by exactly q (see Crossrefs).
The terminology stems from the mapping of the i-th digit onto quantum states of the i-th particle (-1/2 for digit '0', +1/2 for digit '1'), the numbers onto quantum states of the system, and the pairs onto quantum transitions between states. In magnetic resonance (NMR) the most intense transitions are the single-quantum ones (q=1) with k=0, called "main transitions", while those with k>0, called "combination transitions", tend to be weaker. Zero-, double- and, in general, q-quantum transitions are detectable by special techniques.
[Specific case]: This sequence is for single-quantum transitions (q = 1). It lists the flattened triangle T(1;N,k), with rows N = 1,2,... and columns k = 0..floor((N-1)/2).

			T(1;3,1) = 3 because the only transitions compatible with q=1,k=1 are (001,110),(010,101),(100,011).
Starting rows of the triangle T(1;N,k):
  N | k = 0, 1, ..., floor((N-1)/2)
  1 |  1
  2 |  4
  3 | 12   3
  4 | 32  24
  5 | 80 120 10

R. R. Ernst, G. Bodenhausen, A. Wokaun, Principles of nuclear magnetic resonance in one and two dimensions, Clarendon Press, 1987, Chapters 2-6.
M. H. Levitt, Spin Dynamics, J.Wiley & Sons, 2nd Ed.2007, Part3 (Section 6).
J. A. Pople, W. G. Schneider, H. J. Bernstein, High-resolution Nuclear Magnetic Resonance, McGraw-Hill, 1959, Chapter 6.

Stanislav Sykora, Table of n, a(n) for n = 1..2550
Stanislav Sykora, T(1;N,k) with rows N=1,..,100 and columns k=0,..,floor((N-1)/2)
S. Sykora, p-Quantum Transitions and a Combinatorial Identity, Stan's Library, II, Aug 2007.
Stanislav Sýkora, Magnetic Resonance on OEIS, Stan's NMR Blog (Dec 31, 2014), Retrieved Nov 12, 2019.

Cf. A051288 (q=0), A213344..A213352 (q=2..10).

Cf. A001787 (first column), A001791 (row sums).

egf := exp(2*x*y) * BesselI(1, 2*x):
ser := series(egf, x, 32): cx := n -> coeff(ser, x, n):
Trow := n -> n!*seq(coeff(cx(n), y, n - 2*k - 1), k = 0..floor((n-1)/2)):
seq(print([n], Trow(n)), n = 1..12); # Peter Luschny, May 12 2021

With[{q = 1}, Table[2^(n - q - 2 k)*Binomial[n, k] Binomial[n - k, q + k], {n, 11}, {k, 0, Floor[(n - 1)/2]}]] // Flatten (* Michael De Vlieger, Nov 18 2019 *)

TNQK(N, q, k)={binomial(N, k)*binomial(N-k, q+k)*2^((N-k)-(q+k))}
TQ(Nmax, q)={vector(Nmax-q+1, n, vector(1+(n-1)\2, k, TNQK(n+q-1, q, k-1)))}
{ concat(TQ(13, 1)) } \\ simplified by Andrew Howroyd, May 12 2021

Set q = 1 in: T(q;N,k) = 2^(N-q-2*k)*binomial(N,k)*binomial(N-k,q+k).

T(n, k) = n! * [y^(n-2*k-1)] [x^n] exp(2*x*y)*BesselI(1, 2*x). - Peter Luschny, May 12 2021

A345959 Numbers whose prime indices have alternating sum -1.

Original entry on oeis.org

6, 15, 24, 35, 54, 60, 77, 96, 135, 140, 143, 150, 216, 221, 240, 294, 308, 315, 323, 375, 384, 437, 486, 540, 560, 572, 600, 667, 693, 726, 735, 864, 875, 884, 899, 960, 1014, 1147, 1176, 1215, 1232, 1260, 1287, 1292, 1350, 1500, 1517, 1536, 1715, 1734, 1748

Offset: 1

Gus Wiseman, Jul 12 2021

nonn

A prime index of n is a number m such that prime(m) divides n. The multiset of prime indices of n is row n of A112798.
The alternating sum of a sequence (y_1,...,y_k) is Sum_i (-1)^(i-1) y_i. Of course, the alternating sum of prime indices is also the reverse-alternating sum of reversed prime indices.
Also numbers with even Omega (A001222) and exactly one odd conjugate prime index. Conjugate prime indices are listed by A321650, ranked by A122111.

			The initial terms and their prime indices:
    6: {1,2}
   15: {2,3}
   24: {1,1,1,2}
   35: {3,4}
   54: {1,2,2,2}
   60: {1,1,2,3}
   77: {4,5}
   96: {1,1,1,1,1,2}
  135: {2,2,2,3}
  140: {1,1,3,4}
  143: {5,6}
  150: {1,2,3,3}
  216: {1,1,1,2,2,2}
  221: {6,7}
  240: {1,1,1,1,2,3}

These multisets are counted by A000070.
The k = 0 version is A000290, counted by A000041.
The k = 1 version is A001105.
The k > 0 version is A026424.
These are the positions of -1's in A316524.
The k = 2 version is A345960.
The k = -2 version is A345962.
A000984/A345909/A345911 count/rank compositions with alternating sum 1.
A001791/A345910/A345912 count/rank compositions with alternating sum -1.
A027187 counts partitions with reverse-alternating sum <= 0.
A056239 adds up prime indices, row sums of A112798.
A097805 counts compositions by alternating (or reverse-alternating) sum.
A103919 counts partitions by sum and alternating sum (reverse: A344612).
A325534/A325535 count separable/inseparable partitions.
A344607 counts partitions with reverse-alternating sum >= 0.
A344616 gives the alternating sum of reversed prime indices.
Cf. A000097, A028260, A035363, A236913, A239830, A341446, A344609, A344610, A344651, A345919, A345961.

primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
ats[y_]:=Sum[(-1)^(i-1)*y[[i]],{i,Length[y]}];
Select[Range[0,100],ats[primeMS[#]]==-1&]

A345961 Numbers whose prime indices have reverse-alternating sum 2.

Original entry on oeis.org

3, 10, 12, 21, 27, 30, 40, 48, 55, 70, 75, 84, 90, 91, 108, 120, 147, 154, 160, 187, 189, 192, 210, 220, 243, 247, 250, 270, 280, 286, 300, 336, 360, 363, 364, 391, 432, 442, 462, 480, 490, 495, 507, 525, 551, 588, 616, 630, 640, 646, 675, 713, 748, 750, 756

Offset: 1

Gus Wiseman, Jul 12 2021

nonn

A prime index of n is a number m such that prime(m) divides n. The multiset of prime indices of n is row n of A112798.
The reverse-alternating sum of a sequence (y_1,...,y_k) is Sum_i (-1)^(k-i) y_i. Of course, the reverse-alternating sum of prime indices is also the alternating sum of reversed prime indices.
Also numbers with exactly two odd conjugate prime indices. The restriction to odd omega is A345960, and the restriction to even omega is A345962.

			The initial terms and their prime indices:
    3: {2}
   10: {1,3}
   12: {1,1,2}
   21: {2,4}
   27: {2,2,2}
   30: {1,2,3}
   40: {1,1,1,3}
   48: {1,1,1,1,2}
   55: {3,5}
   70: {1,3,4}
   75: {2,3,3}
   84: {1,1,2,4}
   90: {1,2,2,3}
   91: {4,6}
  108: {1,1,2,2,2}
  120: {1,1,1,2,3}

Below we use k to indicate reverse-alternating sum.
The k > 0 version is A000037.
These multisets are counted by A000097.
The k = 0 version is A000290, counted by A000041.
These partitions are counted by A120452 (negative: A344741).
These are the positions of 2's in A344616.
The k = -1 version is A345912.
The k = 1 version is A345958.
The unreversed version is A345960 (negative: A345962).
A000070 counts partitions with alternating sum 1.
A002054/A345924/A345923 count/rank compositions with alternating sum -2.
A027187 counts partitions with reverse-alternating sum <= 0.
A056239 adds up prime indices,  row sums of A112798.
A088218/A345925/A345922 count/rank compositions with alternating sum 2.
A088218 also counts compositions with alternating sum 0, ranked by A344619.
A097805 counts compositions by alternating (or reverse-alternating) sum.
A103919 counts partitions by sum and alternating sum (reverse: A344612).
A316524 gives the alternating sum of prime indices.
A325534 and A325535 count separable and inseparable partitions.
A344606 counts alternating permutations of prime indices.
A344610 counts partitions by sum and positive reverse-alternating sum.
Cf. A000984, A001791, A025047, A027193, A239830, A341446, A344611, A344650, A344651, A344743, A345910, A345911, A345918.

primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
sats[y_]:=Sum[(-1)^(i-Length[y])*y[[i]],{i,Length[y]}];
Select[Range[100],sats[primeMS[#]]==2&]

A007946 a(n) = 6(2n+1)! / ((n!)^2*(n+3)).

Original entry on oeis.org

2, 9, 36, 140, 540, 2079, 8008, 30888, 119340, 461890, 1790712, 6953544, 27041560, 105306075, 410605200, 1602881040, 6263890380, 24502865310, 95937144600, 375945078600, 1474358525640, 5786272150230, 22724268808176, 89301056353200, 351140573438200, 1381487341784004

Offset: 0

David W. Wilson and Dean Hickerson, Apr 21 1997

If Y is a fixed 2-subset of a 2n-set X then a(n-2) is the number of (n-1)-subsets of X intersecting Y. - Milan Janjic, Oct 21 2007

G. C. Greubel, Table of n, a(n) for n = 0..1000
Milan Janjic, Two Enumerative Functions.

Cf. A000120, A001622, A001791, A002054, A061549, A242986.

[Binomial(2*n+2, n) + Binomial(2*n+3, n) : n in [0..30]]; // Wesley Ivan Hurt, Aug 23 2014

A007946:=n->binomial(2*n+2,n)+binomial(2*n+3,n): seq(A007946(n), n=0..30); # Wesley Ivan Hurt, Aug 23 2014

Table[Binomial[2 n + 2, n] + Binomial[2 n + 3, n], {n, 0, 30}] (* Wesley Ivan Hurt, Aug 23 2014 *)
Table[6*(2*n + 1)!/((n!)^2*(n + 3)), {n,0,50}] (* G. C. Greubel, Jan 23 2017 *)

for(n=0,50, print1(6*(2*n + 1)!/((n!)^2*(n + 3)), ", ")) \\ G. C. Greubel, Jan 23 2017

a(n) = C(2n+2, n) + C(2n+3, n). - Emeric Deutsch, May 16 2003
From Karol A. Penson, Aug 23 2014: (Start)
O.g.f.: ((-2+1/z^2-2/z)/sqrt(1-4*z)-1/z^2)/(2*z).
Representation as the n-th moment of a signed function: w(x) = sqrt(x/(4-x))*(x^2-2*x-2)/(2*Pi) on the segment x = (0,4): a(n) = Integral_{x=0..4} x^n*w(x) dx. For x->0, w(x)->0, and for x->4, w(x)->infinity.
a(n) ~ (3/65536)*(4^n)*(-55332459+18443992*n - 6147840*n^2 + 2050048*n^3 - 688128*n^4 + 262144*n^5)/(n^(11/2)*sqrt(Pi)), for n->infinity.
(End)
a(n) = A001791(n+1) + A002054(n+1). - Wesley Ivan Hurt, Aug 23 2014
From Peter Luschny, Aug 25 2014: (Start)
a(n) = ((6*(2*n+1))/(n+3))* binomial(2*n,n).
a(n) has the asymptotic series 2^(2*n+3)*(1+(n+3)/((2*n+3))) *Sum_{k>=0}((num(k)/den(k))*(-n)^(-k))/sqrt(n*Pi). Here den(n) = 2^(4*n-A000120(n)) = A061549(n) and num(n) = 1, 25, 1297, 32755, 3249099, 79652055, 3876842453, 93900904955, 18138634602803, 437081823058595, 21036073578365391,... For example a(100) = 0.10602088220899083... *10^61 with the given values of num.
a(x) ~ exp(x*log(4)-(log(Pi)+cos(2*Pi*x)*(log(x) + 1/(4*x)))/2 + log((12*x+6)/ (3+x))). For example, this formula gives a(100) = 0.10602088... *10^61.
a(n) = A242986(2*n). (End)
a(n) = 12*4^n*Gamma(3/2+n)/(sqrt(Pi)*(3+n)*Gamma(1+n)). - Peter Luschny, Dec 14 2015
a(n) = 2*Sum_{i=0..n} (1/(i+1)*binomial(2*i+3,i+3)*binomial(2*(n-i),n-i)). - Vladimir Kruchinin, Apr 20 2016
E.g.f.: 2*(x*(-1 + 3*x)*BesselI(0,2*x) + (1 - 2*x + 3*x^2) * BesselI(1,2*x))*exp(2*x)/x^2. - Ilya Gutkovskiy, Apr 20 2016
D-finite with recurrence n*(n+3)*a(n) -2*(n+2)*(2*n+1)*a(n-1) = 0. - R. J. Mathar, Mar 30 2022
From Amiram Eldar, Feb 16 2023: (Start)
Sum_{n>=0} 1/a(n) = 8*Pi/(27*sqrt(3)) + 1/9.
Sum_{n>=0} (-1)^n/a(n) = 8*log(phi)/(5*sqrt(5)) + 1/15, where phi is the golden ratio (A001622). (End)

A108044 Triangle read by rows: right half of Pascal's triangle (A007318) interspersed with 0's.

Original entry on oeis.org

1, 0, 1, 2, 0, 1, 0, 3, 0, 1, 6, 0, 4, 0, 1, 0, 10, 0, 5, 0, 1, 20, 0, 15, 0, 6, 0, 1, 0, 35, 0, 21, 0, 7, 0, 1, 70, 0, 56, 0, 28, 0, 8, 0, 1, 0, 126, 0, 84, 0, 36, 0, 9, 0, 1, 252, 0, 210, 0, 120, 0, 45, 0, 10, 0, 1, 0, 462, 0, 330, 0, 165, 0, 55, 0, 11, 0, 1, 924, 0, 792, 0, 495, 0, 220, 0, 66

Offset: 0

N. J. A. Sloane, Jun 02 2005

Column k has e.g.f. Bessel_I(k,2x). - Paul Barry, Mar 10 2010
T(n,k) is the number of binary sequences of length n in which the number of ones minus the number of zeros is k. If 2 divides(n+k), such a sequence will have (n+k)/2 ones and (n-k)/2 zeros. Since there are C(n,(n+k)/2) ways to choose the sequence entries that get a one, T(n,k)=binomial(n,(n+k)/2) whenever (n+k) is even and T(n,k)= 0 otherwise. See the example below in the example section. - Dennis P. Walsh, Apr 11 2012
T(n,k) is the number of walks on the semi-infinite integer line with n steps that end at k. The walks start at 0, move at each step either one to the left or one to the right, and never enter the region of negative k. [Walks with impenetrable wall at -1/2. Dyck excursions of n steps that end at level k.] The variant without the restriction of negative positions is A053121. - R. J. Mathar, Nov 02 2023

			Triangle begins:
  1
  0 1
  2 0 1
  0 3 0 1
  6 0 4 0 1
  0 10 0 5 0 1
  20 0 15 0 6 0 1
From _Paul Barry_, Mar 10 2010: (Start)
Production matrix is
  0, 1,
  2, 0, 1,
  0, 1, 0, 1,
  0, 0, 1, 0, 1,
  0, 0, 0, 1, 0, 1,
  0, 0, 0, 0, 1, 0, 1,
  0, 0, 0, 0, 0, 1, 0, 1,
  0, 0, 0, 0, 0, 0, 1, 0, 1,
  0, 0, 0, 0, 0, 0, 0, 1, 0, 1 (End)
T(5,1)=10 since there are 10 binary sequences of length 5 in which the number of ones exceed the number of zeros by exactly 1, namely, 00111, 01011, 01101, 01110, 10011, 10101, 10110, 11001, 11010, and 11100. Also, T(5,2)=0 since there are no binary sequences in which the number of ones exceed the number of zeros by exactly 2. - _Dennis P. Walsh_, Apr 11 2012

Harvey P. Dale, Rows n = 0..125 of triangle, flattened
Paul Barry and Aoife Hennessy, Meixner-Type Results for Riordan Arrays and Associated Integer Sequences, J. Int. Seq. 13 (2010) # 10.9.4, section 8.
Johann Cigler, Some elementary observations on Narayana polynomials and related topics, arXiv:1611.05252 [math.CO], 2016. See p. 14.
Robert S. Maier, Sheffer Polynomials and the s-ordering of Exponential Boson Operators, arXiv:2508.13094 [quant-ph], 2025. See p. 27.
Paul Peart and Wen-Jin Woan, A divisibility property for a subgroup of Riordan matrices, Discrete Applied Mathematics, Vol. 98, Issue 3, Jan 2000, 255-263.
Louis W. Shapiro, Seyoum Getu, Wen-Jin Woan, and Leon C. Woodson, The Riordan Group, Discrete Appl. Maths. 34 (1991) 229-239.
Index entries for triangles and arrays related to Pascal's triangle

Cf. A007318, A108045 (matrix inverse),

Cf. A204293, A357136, A000984 (column 0), A001700 (column 1), A001791 (column 2), A002054 (column 3)

import Data.List (intersperse)
a108044 n k = a108044_tabl !! n !! k
a108044_row n = a108044_tabl !! n
a108044_tabl = zipWith drop [0..] $ map (intersperse 0) a007318_tabl
-- Reinhard Zumkeller, May 18 2013

T:=proc(n,k) if n+k mod 2 = 0 then binomial(n,(n+k)/2) else 0 fi end: for n from 0 to 13 do seq(T(n,k),k=0..n) od; # yields sequence in triangular form; Emeric Deutsch, Jun 05 2005

b[n_,k_]:=If[EvenQ[n+k],Binomial[n,(n+k)/2],0]; Flatten[Table[b[n,k],{n,0,20},{k,0,n}]] (* Harvey P. Dale, May 05 2013 *)

Each entry is the sum of those in the previous row that are to its left and to its right.
Riordan array (1/sqrt(1-4*x^2), (1-sqrt(1-4*x^2))/(2*x)).
T(n, k) = binomial(n, (n+k)/2) if n+k is even, T(n, k)=0 if n+k is odd. G.f.=f/(1-tg), where f=1/sqrt(1-4x^2) and g=(1-sqrt(1-4x^2))/(2x). - Emeric Deutsch, Jun 05 2005
From Peter Bala, Jun 29 2015: (Start)
Riordan array has the form ( x*h'(x)/h(x), h(x) ) with h(x) = ( 1 - sqrt(1 - 4*x) )/(2*x) and so belongs to the hitting time subgroup H of the Riordan group (see Peart and Woan).
T(n,k) = [x^(n-k)] f(x)^n with f(x) = 1 + x^2. In general the (n,k)th entry of the hitting time array ( x*h'(x)/h(x), h(x) ) has the form [x^(n-k)] f(x)^n, where f(x) = x/( series reversion of h(x) ).
The inverse array is A108045 (a hitting time array with h(x) = x/(1 + x^2)). (End)

More terms from Emeric Deutsch and Christian G. Bower, Jun 05 2005

A132812 Triangle read by rows, n>=1, 1<=k<=n, T(n,k) = k*binomial(n,k)^2/(n-k+1).

Original entry on oeis.org

1, 2, 2, 3, 9, 3, 4, 24, 24, 4, 5, 50, 100, 50, 5, 6, 90, 300, 300, 90, 6, 7, 147, 735, 1225, 735, 147, 7, 8, 224, 1568, 3920, 3920, 1568, 224, 8, 9, 324, 3024, 10584, 15876, 10584, 3024, 324, 9, 10, 450, 5400, 25200, 52920, 52920, 25200, 5400, 450, 10

Offset: 1

Gary W. Adamson, Sep 01 2007

A127648 * A001263. (Original name by Gary W. Adamson.)
Let a meander be defined as in the link and m = 2. Then T(n,k) counts the invertible meanders of length m(n+1) built from arcs with central angle 360/m whose binary representation have mk '1's. - Peter Luschny, Dec 19 2011
Antidiagonal sums = A110320. - Philippe Deléham, Jun 08 2013

			First few rows of the triangle are:
  1;
  2, 2;
  3, 9, 3;
  4, 24, 24, 4;
  5, 50, 100, 50, 5;
  6, 90, 300, 300, 90, 6;
  ...
Row 4 = (4, 24, 24, 4) = 4 * (1, 6, 6, 1), where (1, 6, 6, 1) = row 4 of the Narayana triangle. - _Gary W. Adamson_
T(3,1) = 3 because the invertible meanders of length 8 and central angle 180 degree which have two '1's in their binary representation are {10000100, 10010000, 11000000}. - _Peter Luschny_, Dec 19 2011

Michael De Vlieger, Table of n, a(n) for n = 1..11325 (rows 1..150).
Peter Luschny, Meanders.

Row sums: A001791. Cf. A001263, A127648, A001791, A202409.

A132812 := (n,k) -> k*binomial(n,k)^2/(n-k+1);
seq(print(seq(A132812(n,k),k=0..n-1)),n=1..6); # Peter Luschny, Dec 19 2011

Table[k Binomial[n, k]^2/(n - k + 1), {n, 10}, {k, n}] // Flatten (* Michael De Vlieger, Nov 15 2017 *)

A127648 * A001263 as infinite lower triangular matrices.
a(n) = n * A001263(n,k).
T(n,k) = binomial(n,k)*binomial(n,k-1). - Philippe Deléham, Jun 08 2013
G.f.: x*d(N(x,y))/dx, where N(x,y) is g.f. for Narayana numbers A001263. - Vladimir Kruchinin, Oct 22 2021

New name from Peter Luschny, Dec 19 2011

a(53) corrected by Michael De Vlieger, Nov 15 2017

cp's OEIS Frontend

A000917 a(n) = (2n+3)!/(n!*(n+2)!).

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A025566 a(n) = number of (s(0), s(1), ..., s(n)) such that every s(i) is a nonnegative integer, s(0) = 0, s(1) = 1, |s(i) - s(i-1)| <= 1 for i >= 2. Also a(n) = sum of numbers in row n+1 of the array T defined in A026105. Also a(n) = T(n,n), where T is the array defined in A025564.

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A029760 A sum with next-to-central binomial coefficients of even order, Catalan related.

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A100824 Number of partitions of n with at most one odd part.

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A213343 1-quantum transitions in systems of N spin 1/2 particles, in columns by combination indices. Triangle read by rows, T(n, k) for n >= 1 and 0 <= k <= floor((n-1)/2).

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A345959 Numbers whose prime indices have alternating sum -1.

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A345961 Numbers whose prime indices have reverse-alternating sum 2.

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A007946 a(n) = 6(2n+1)! / ((n!)^2*(n+3)).