A005430 -id:A005430 - cp's OEIS frontend

A097070 Consider all compositions (ordered partitions) of n into n parts, allowing zeros. E.g., for n = 3 we get 300, 030, 003, 210, 120, 201, 102, 021, 012, 111. Then a(n) is the total number of 1's.

1, 2, 9, 40, 175, 756, 3234, 13728, 57915, 243100, 1016158, 4232592, 17577014, 72804200, 300874500, 1240940160, 5109183315, 21002455980, 86213785350, 353452638000, 1447388552610, 5920836618840, 24197138082780, 98801168731200, 403095046038750, 1643337883690776, 6694900194799404

Offset: 1

Amy J. Kolan, Sep 15 2004

Number of compositions of n into n parts, allowing zeros = binomial(2*n-1,n) = A088218 = essentially A001700.

			The compositions for n=2 are 20, 02, 11. There are two 1's in these so a(2) = 2.
From _Robert G. Wilson v_, Sep 16 2004: (Start)
The case n = 5:
A. There are 5 combinations associated with the numbers 50000: 50000, 05000, 00500, 00050, 00005.
B. There are 20 combinations associated with the numbers 41000.
C. There are 20 combinations associated with 32000.
D. There are 30 combinations associated with 31100.
E. There are 30 combinations associated with 22100.
F. There are 20 combinations associated with 21110.
G. There is one combinations associated with 11111.
The number of 1's associated with A is 0, with B 20, with C 0, with D 60, with E 30, with F 60 and with G 5. 0 + 20 + 0 + 60 + 30 + 60 + 5 = 175.
(End)

Vincenzo Librandi, Table of n, a(n) for n = 1..1000
Paul Barry, On a Central Transform of Integer Sequences, arXiv:2004.04577 [math.CO], 2020.
G.-S. Cheon, H. Kim, and L. W. Shapiro, Mutation effects in ordered trees, arXiv preprint arXiv:1410.1249 [math.CO], 2014.

Cf. A037965.
Cf. A000984, A001622, A088218.
cf. A005430, A002740, A002457, A000917, A110609, A097613, A001791, A110609.

List([1..30], n-> n*Binomial(2*n-3, n-1)); # G. C. Greubel, Jul 27 2019

[n*Binomial(2*n-3, n-1): n in [1..30]]; // Vincenzo Librandi, Jul 13 2019

A097070 := n -> ifelse(n=1, 1, 2^(n-2)*JacobiP(n-1, -1/2, -n+2, 3)):
seq(simplify(A097070(n)), n = 1..28);  # Peter Luschny, Jan 22 2025

Table[n*Binomial[2n-3, n-1], {n, 30}] (* Robert G. Wilson v, Sep 17 2004 *)

a(n) = n*binomial(2*n-3, n-1); \\ Joerg Arndt, Feb 17 2015

[n*binomial(2*n-3, n-1) for n in (1..30)] # G. C. Greubel, Jul 27 2019

a(n) = n*binomial(2*n-3, n-1).
More generally, total number of k's (k>=0) in all ordered partitions of n into n parts, allowing zeros, is n*binomial(2*n-k-2, n-2) if n >= k, 0 otherwise.
Total number of 0's is given by A005430.
From Vladeta Jovovic, Sep 17 2004: (Start)
a(n) = Sum_{k=0..n} k*binomial(n, k)*binomial(n-2, k-2).
G.f.: x*(1 -2*x +(1-4*x)^(3/2))/(2*(1-4*x)^(3/2)).
E.g.f.: (x/2)*(exp(2*x)*BesselI(0, 2*x)+1). (End)
a(n) = A014107(n)*A000108(n-2). - Philippe Deléham, Apr 12 2007
a(n) = n*A088218(n-1) for n > 0. - Werner Schulte, Jan 22 2017
From Bruce J. Nicholson, Jul 11 2019: (Start)
a(n) = A002740(n) + A097613(n).
a(n) = A110609(n-1) - A002457(n-2) + A097613(n).
a(n) = A005430(n-1) - A000917(n-3) for n > 1.
a(n) = A002457(n-1) - A037965(n) - A000917(n-3) for n > 1.
a(n) = A037965(n)/2.
a(n) = A001700(n-2)*n.
a(n) = A001791(n-2)*n + A000984(n-2)*n for n > 1. (End)
From Amiram Eldar, May 16 2022: (Start)
Sum_{n>=1} 1/a(n) = 4*Pi/(3*sqrt(3)) - Pi^2/9.
Sum_{n>=1} (-1)^(n+1)/a(n) = 8*log(phi)/sqrt(5) - 4*log(phi)^2, where phi is the golden ratio (A001622). (End)
a(n) = 2^(n-2)*JacobiP(n-1, -1/2, -n+2, 3) for n > 1. - Peter Luschny, Jan 22 2025

Formula, more terms and comments from Vladeta Jovovic, Sep 15 2004

A002011 a(n) = 4*(2n+1)!/n!^2.

Original entry on oeis.org

4, 24, 120, 560, 2520, 11088, 48048, 205920, 875160, 3695120, 15519504, 64899744, 270415600, 1123264800, 4653525600, 19234572480, 79342611480, 326704870800, 1343120024400, 5513861152800, 22606830726480, 92580354403680, 378737813469600

Offset: 0

N. J. A. Sloane, Simon Plouffe

nonn

R. C. Mullin, E. Nemeth and P. J. Schellenberg, The enumeration of almost cubic maps, pp. 281-295 in Proceedings of the Louisiana Conference on Combinatorics, Graph Theory and Computer Science. Vol. 1, edited R. C. Mullin et al., 1970.
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
Milan Janjić, Pascal Matrices and Restricted Words, J. Int. Seq., Vol. 21 (2018), Article 18.5.2.

a(n)=4 A002457(n).
a(n) = 2 * A005430(n+1) = 4 * A002457(n).
Cf. A001803.

seq(2*n*binomial(2*n,n), n=1..23); # Zerinvary Lajos, Dec 14 2007

Table[4*(2*n + 1)!/n!^2, {n, 0, 20}] (* T. D. Noe, Aug 30 2012 *)

PARI
```
a(n)=if(n<0,0,4*(2*n+1)!/n!^2)
```

G.f.: 4*(1-4x)^(-3/2).

a(n) = 1/J(n) where J(n) = Integral_{t=0..Pi/4} (cos(t)^2 - 1/2)^(2n+1). - Benoit Cloitre, Oct 17 2006

Simpler description from Travis Kowalski (tkowalski(AT)coloradocollege.edu), Mar 20 2003

A061928 Array T(n,m) = 1/beta(n+1,m+1) read by antidiagonals.

Original entry on oeis.org

6, 12, 12, 20, 30, 20, 30, 60, 60, 30, 42, 105, 140, 105, 42, 56, 168, 280, 280, 168, 56, 72, 252, 504, 630, 504, 252, 72, 90, 360, 840, 1260, 1260, 840, 360, 90, 110, 495, 1320, 2310, 2772, 2310, 1320, 495, 110, 132, 660, 1980, 3960, 5544, 5544, 3960

Offset: 1

Frank Ellermann, May 22 2001

beta(n+1,m+1) = Integral_{x=0..1} x^n * (1-x)^m dx for real n, m.

			Antidiagonals:
   6,
  12, 12,
  20, 30, 20,
  30, 60, 60, 30,
  ...
Array:
   6  12  20   30   42
  12  30  60  105  168
  20  60 140  280  504
  30 105 280  630 1260
  42 168 504 1260 2772

G. Boole, A Treatise On The Calculus of Finite Differences, Dover, 1960, p. 26.

Rows: 1/b(n, 2): A002378, 1/b(n, 3): A027480, 1/b(n, 4): A033488. Diagonals: 1/b(n, n): A002457, 1/b(n, n+1) A005430, 1/b(n, n+2): A000917.

T(i, j)=A003506(i+1, j+1).

t[n_, m_] := 1/Beta[n+1, m+1]; Take[ Flatten[ Table[ t[n+1-m, m], {n, 1, 10}, {m, 1, n}]], 52] (* Jean-François Alcover, Oct 11 2011 *)

A(i,j)=if(i<1||j<1,0,1/subst(intformal(x^i*(1-x)^j),x,1)) /* Michael Somos, Feb 05 2004 */

A(i,j)=if(i<1||j<1,0,1/sum(k=0,i,(-1)^k*binomial(i,k)/(j+1+k))) /* Michael Somos, Feb 05 2004 */

from sympy import factorial as f
def T(n, m): return f(n + m + 1)/(f(n)*f(m))
for n in range(1, 11): print([T(m, n - m + 1) for m in range(1, n + 1)]) # Indranil Ghosh, Apr 29 2017

beta(n+1, m+1) = gamma(n+1)*gamma(m+1)/gamma(n+m+2) = n!*m!/(n+m+1)!.

A212303 a(n) = n!/([(n-1)/2]!*[(n+1)/2]!) for n>0, a(0)=0, and where [ ] = floor.

Original entry on oeis.org

0, 1, 2, 3, 12, 10, 60, 35, 280, 126, 1260, 462, 5544, 1716, 24024, 6435, 102960, 24310, 437580, 92378, 1847560, 352716, 7759752, 1352078, 32449872, 5200300, 135207800, 20058300, 561632400, 77558760, 2326762800, 300540195, 9617286240, 1166803110, 39671305740

Offset: 0

Peter Luschny, Oct 24 2013

nonn

a(n) + A056040(n) = A189911(n), the row sums of the extended Catalan triangle A189231.

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Peter Luschny, The lost Catalan numbers.

Cf. A005430, A001700, A056040, A189911.

A212303 := proc(n) if n mod 2 = 0 then n*binomial(n, iquo(n,2))/2 else binomial(n+1, iquo(n,2)+1)/2 fi end: seq(A212303(i), i=0..36);

a[n_?EvenQ] := n*Binomial[n, n/2]/2; a[n_?OddQ] := Binomial[n+1, Quotient[n, 2]+1]/2; Table[a[n], {n, 0, 36}]  (* Jean-François Alcover, Feb 05 2014 *)
nxt[{n_,a_}]:={n+1,If[OddQ[n],a(n+1),(4a(n+1))/(n(n+2))]}; Join[{0}, Transpose[ NestList[ nxt,{1,1},40]][[2]]] (* Harvey P. Dale, Dec 20 2014 *)

def A212303():
    yield 0
    r, n = 1, 1
    while True:
        yield r
        n += 1
        r *= n if is_even(n) else 4*n/((n-1)*(n+1))
a = A212303(); [next(a) for i in range(36)]

E.g.f.: (1+x)*BesselI(1, 2*x).
O.g.f.: -((4*x^2-1)^(3/2)+I-(4*I)*x^2+(4*I)*x^3)/(2*x*(4*x^2-1)^(3/2)).
Recurrence: a(n) = n if n < 2 else a(n) = a(n-1)*n if n is even else a(n-1)*n*4/((n-1)*(n+1)).
a(2*n) = n*C(2*n, n) (A005430); a(2*n+1) = C(2*n+1,  n+1) (A001700).
a(n) = n$*floor((n+1)/2)^((-1)^n), where n$ is the swinging factorial A056040.
a(n) = Sum_{k=0..n} A189231(n, 2*k+1).
Sum_{n>=1} 1/a(n) = 2/3 + (7/27)*sqrt(3)*Pi.
Sum_{n>=1} (-1)^(n+1)/a(n) = 2/3 + Pi/(9*sqrt(3)). - Amiram Eldar, Aug 20 2022

A173384 a(n) = 2^(2n - HammingWeight(n)) [x^n] ((x-1)^(-1) + (1-x)^(-3/2)).

Original entry on oeis.org

0, 1, 7, 19, 187, 437, 1979, 4387, 76627, 165409, 707825, 1503829, 12706671, 26713417, 111868243, 233431331, 7770342787, 16124087129, 66765132341, 137948422657, 1138049013461, 2343380261227, 9636533415373, 19787656251221

Offset: 0

Paul Curtz, Feb 17 2010

nonn

If n >= 1 it appears a(n-1) is equal to the difference between the denominator and the numerator of the ratio (2n-1)!!/(2n-2)!!. In particular a(n-1) is the difference between the denominator and the numerator of the ratio A001147(2n-2)/A000165(2n-1). See examples. - Anthony Hernandez, Feb 05 2020
It can be seen that this is true, e.g., using A001803(n) = (2n+1)!/(n!^2*2^A000120(n)) and A046161(n) = 4^n/2^A000120(n). - M. F. Hasler, Feb 07 2020
Numerators in the expansion of (1-(1-x)^(1/2))/(1-x)^(3/2). Denominators are A046161. Compare to A001790. - Thomas Curtright, Feb 09 2024

			From _Anthony Hernandez_, Feb 05 2020: (Start)
Consider n = 4. The 4th odd number is 7, and 7!!/(7-1)!! = 35/16, and a(4-1) = a(3) = 35 - 16 = 19.
Consider n = 7. The 7th odd number is 13, and 13!!/(13-1)!! = 3003/1024, and a(7-1) = a(6) = 3003 - 1024 = 1979. (End)

G. C. Greubel, Table of n, a(n) for n = 0..1000
Thomas Curtright and Gaurav Verma, Scattering Shadows, arXiv:2404.07745 [physics.class-ph], 2024, p. 9.

Cf. A001790, A005430, A046161.

List([0..30], n-> (NumeratorRat((2*n+1)*Binomial(2*n, n)/(4^n)) - DenominatorRat(Binomial(2*n, n)/(4^n)))); # G. C. Greubel, Dec 09 2018

[Numerator((2*n+1)*Binomial(2*n, n)/(4^n)) - Denominator(Binomial(2*n, n)/(4^n)): n in [0..30]]; // G. C. Greubel, Dec 09 2018

A046161 := proc(n) binomial(2*n,n)/4^n ; denom(%) ; end proc:
A173384 := proc(n) A001803(n)-A046161(n) ; end proc: # R. J. Mathar, Jul 06 2011

Table[Numerator[(2*n+1)*Binomial[2*n, n]/(4^n)] - Denominator[Binomial[2*n, n]/(4^n)], {n,0,30}] (* G. C. Greubel, Dec 09 2018 *)
A173384[n_] := 2^(2*n - DigitCount[n, 2, 1]) Coefficient[Series[(x - 1)^(-1) + (1 - x)^(-3/2), {x, 0, n}], x, n]
Table[A173384[n], {n, 0, 23}]  (* Peter Luschny, Feb 17 2024 *)

for(n=0,30, print1(numerator((2*n+1)*binomial(2*n, n)/(4^n)) - denominator(binomial(2*n, n)/4^n), ", ")) \\ G. C. Greubel, Dec 09 2018

[(numerator((2*n+1)*binomial(2*n, n)/(4^n)) - denominator(binomial(2*n, n)/(4^n))) for n in range(30)] # G. C. Greubel, Dec 09 2018

a(n) = A001803(n) - A046161(n). (Previous name.)

Let r(n) = (-2)^n*Sum_{j=0..n-1} binomial(n,j)*Bernoulli(j+n+1, 1/2)/(j+n+1) then a(n) = numerator(r(n)). - Peter Luschny, Jun 20 2017

New name using an expansion of Thomas Curtright by Peter Luschny, Feb 17 2024

A241519 Denominators of b(n) = b(n-1)/2 + 1/(2*n), b(0)=0.

Original entry on oeis.org

1, 2, 2, 12, 3, 15, 60, 840, 105, 630, 630, 13860, 6930, 180180, 360360, 144144, 9009, 306306, 306306, 11639628, 14549535, 14549535, 58198140, 2677114440, 334639305, 3346393050

Offset: 0

Paul Curtz, Apr 24 2014

nonn

Generally, 2*b(n) = b(n-1) + f(n). See, for f(n)=n, A000337(n)/2^n.
a(0)=1. b(n) is mentioned in A241269.
Difference table of b(n):
     0,   1/2,   1/2,  5/12,    1/3,   4/15, ...
   1/2,     0, -1/12, -1/12,  -1/15,  -1/20, ...
  -1/2, -1/12,     0,  1/60,   1/60, 11/840, ...
  5/12,  1/12,  1/60,     0, -1/280, -1/280, ...
etc.
b(n) is mentioned in A241269 as an autosequence of the first kind.
The denominators of the first two upper diagonals are the positive Apéry numbers, A005430(n+1). Compare to the array in A003506.
Numerators: 0, 1, 1, 5, 1, 4, 13, 151, 16, 83, 73, 1433, 647, 15341, ... .

			0, 1/2, 1/2, 5/12, 1/3, 4/15, 13/60, 151/840, 16/105, 83/630, 73/630, ...
b(1) = (0+1)/2, hence a(1)=2.
b(2) = (1/2+1/2)/2 = 1/2, hence a(2)=2.
b(3) = (1/2+1/3)/2 = 5/12, hence a(3)=12.

Cf. A086466.

Cf. A242376 (numerators).

b[0] = 0; b[n_] := b[n] = 1/2*(b[n-1] + 1/n); Table[b[n] // Denominator, {n, 0, 25}] (* Jean-François Alcover, Apr 25 2014 *)
Table[-Re[LerchPhi[2, 1, n + 1]], {n, 0, 20}] // Denominator (* Eric W. Weisstein, Dec 11 2017 *)
-Re[LerchPhi[2, 1, Range[20]]] // Denominator (* Eric W. Weisstein, Dec 11 2017 *)
RecurrenceTable[{b[n] == b[n - 1]/2 + 1/(2 n), b[0] == 0}, b[n], {n, 20}] // Denominator (* Eric W. Weisstein, Dec 11 2017 *)

b(n) = -Re(Phi(2, 1, n + 1)) where Phi denotes the Lerch transcendent. - Eric W. Weisstein, Dec 11 2017

Extension, after a(13), from Jean-François Alcover, Apr 24 2014

A344108 Expansion of Product_{k>=1} 1 / (1 - x^k)^binomial(2*k,k).

Original entry on oeis.org

1, 2, 9, 36, 154, 644, 2744, 11608, 49267, 208610, 882963, 3731640, 15754327, 66426946, 279766063, 1176920484, 4945739292, 20761707824, 87069433162, 364802647912, 1527072152856, 6386873581244, 26690795165394, 111453873957936, 465055114353616, 1939114409985956

Offset: 0

Ilya Gutkovskiy, May 09 2021

nonn

Euler transform of A000984.

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

Cf. A000984, A005430, A088327, A194353, A344130.

a:= proc(n) option remember; `if`(n=0, 1, add(a(n-j)*add(d*
      binomial(2*d, d), d=numtheory[divisors](j)), j=1..n)/n)
    end:
seq(a(n), n=0..25);  # Alois P. Heinz, Feb 13 2023

nmax = 25; CoefficientList[Series[Product[1/(1 - x^k)^Binomial[2 k, k], {k, 1, nmax}], {x, 0, nmax}], x]
a[n_] := a[n] = If[n == 0, 1, (1/n) Sum[Sum[d Binomial[2 d, d], {d, Divisors[k]}] a[n - k], {k, 1, n}]]; Table[a[n], {n, 0, 25}]
nmax = 25; CoefficientList[Series[Exp[Sum[(1/Sqrt[1 - 4*x^j] - 1)/j, {j, 1, nmax}]], {x, 0, nmax}], x] (* Vaclav Kotesovec, May 10 2021 *)

a(0) = 1; a(n) = (1/n) * Sum_{k=1..n} ( Sum_{d|k} A005430(d) ) * a(n-k).

a(n) ~ 2^(2*n - 1/3) * exp(3*n^(1/3)/2^(2/3) - 1 + c) / (sqrt(3*Pi) * n^(5/6)), where c = Sum_{k>=2} (1/sqrt(1 - 4^(1-k)) - 1)/k = 0.0907540019413286886324751305813463657179452545... - Vaclav Kotesovec, May 10 2021

A152659 Triangle read by rows: T(n,k) is the number of lattice paths from (0,0) to (n,n) with steps E=(1,0) and N=(0,1) and having k turns (NE or EN) (1<=k<=2n-1).

Original entry on oeis.org

2, 2, 2, 2, 2, 4, 8, 4, 2, 2, 6, 18, 18, 18, 6, 2, 2, 8, 32, 48, 72, 48, 32, 8, 2, 2, 10, 50, 100, 200, 200, 200, 100, 50, 10, 2, 2, 12, 72, 180, 450, 600, 800, 600, 450, 180, 72, 12, 2, 2, 14, 98, 294, 882, 1470, 2450, 2450, 2450, 1470, 882, 294, 98, 14, 2, 2, 16, 128, 448

Offset: 1

Emeric Deutsch, Dec 10 2008

Row n has 2n-1 entries.
Sum of entries of row n = binomial(2n,n) = A000984(n) (the central binomial coefficients).
Sum(k*T(n,k),k=0..2n-1) = n*binomial(2n,n) = A005430(n).

			T(3,2)=4 because we have ENNNEE, EENNNE, NEEENN and NNEEEN.
Triangle starts:
  2;
  2,2,2;
  2,4,8,4,2;
  2,6,18,18,18,6,2;
  2,8,32,48,72,48,32,8,2;
  ...

Kent E. Morrison, Matching adjacent cards, arXiv:2501.07753 [math.CO], 2025. See alpha_r p. 15.

Cf. A000984, A005430.

T := proc (n, k) if `mod`(k, 2) = 0 then 2*binomial(n-1, (1/2)*k-1)*binomial(n-1, (1/2)*k) else 2*binomial(n-1, (1/2)*k-1/2)^2 end if end proc: for n to 9 do seq(T(n, k), k = 1 .. 2*n-1) end do; # yields sequence in triangular form

T(n,2k) = 2*binomial(n-1,k-1)*binomial(n-1,k);
T(n,2k-1) = 2*binomial(n-1,k-1)^2.
G.f.: [1+t*r(t^2,z)]/[1-t*r(t^2,z)], where r(t,z) is the Narayana function, defined by r = z(1+r)(1+tr).

A373865 Sum over all compositions of 2n into n parts of the least common multiple of the parts.

Original entry on oeis.org

1, 2, 8, 50, 160, 892, 3794, 19560, 80112, 371948, 1614598, 7180494, 30794746, 134269410, 574754496, 2471353090, 10542096528, 45057428356, 191653403306, 814082549052, 3444043955350, 14537736838038, 61174002263338, 256838845740468, 1075631257186986

Offset: 0

Alois P. Heinz, Jun 19 2024

nonn

			a(3) = 50 = 2 + 6*6 + 3*4: 222, 123, 132, 213, 231, 312, 321, 114, 141, 411.

Alois P. Heinz, Table of n, a(n) for n = 0..75

Cf. A005430 (the same for sum), A165817 (the same for product), A181851, A260878 (the same for gcd).

a(n) = A181851(2n,n).

A068554 a(n) = n*binomial(2n, n) - 4^n.

Original entry on oeis.org

-1, -2, -4, -4, 24, 236, 1448, 7640, 37424, 175436, 798984, 3565448, 15672656, 68098936, 293196944, 1253020976, 5322318944, 22491436556, 94632958664, 396682105256, 1657418948624, 6905368852136, 28697991157424, 119000162557136

Offset: 0

N. J. A. Sloane, Mar 23 2002

sign

Known to be >= 0 for n>3.

Hojoo Lee, Posting to Number Theory List, Feb 18 2002.

Harvey P. Dale, Table of n, a(n) for n = 0..1000

Cf. A000302, A005430.

seq(n*binomial(2*n,n)-4^n,n=0..40); # Robert Israel, Nov 13 2016

Table[n*Binomial[2n,n]-4^n,{n,0,30}] (* Harvey P. Dale, Nov 17 2012 *)

From Robert Israel, Nov 13 2016: (Start)
a(n) = A005430(n) - A000302(n).
G.f.: (2*x-sqrt(1-4*x))/(1-4*x)^(3/2).
a(n) = ((16*(n-2))*(2*n-5)*a(n-3)-(4*(8*n^2-23*n+18))*a(n-2)+(2*(5*n-4))*(n-1)*a(n-1))/(n*(n-1)). (End)

cp's OEIS Frontend

A097070 Consider all compositions (ordered partitions) of n into n parts, allowing zeros. E.g., for n = 3 we get 300, 030, 003, 210, 120, 201, 102, 021, 012, 111. Then a(n) is the total number of 1's.

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A002011 a(n) = 4*(2n+1)!/n!^2.

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A061928 Array T(n,m) = 1/beta(n+1,m+1) read by antidiagonals.

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A212303 a(n) = n!/([(n-1)/2]!*[(n+1)/2]!) for n>0, a(0)=0, and where [ ] = floor.

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A173384 a(n) = 2^(2*n - HammingWeight(n)) * [x^n] ((x-1)^(-1) + (1-x)^(-3/2)).

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A241519 Denominators of b(n) = b(n-1)/2 + 1/(2*n), b(0)=0.

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A173384 a(n) = 2^(2n - HammingWeight(n)) [x^n] ((x-1)^(-1) + (1-x)^(-3/2)).