A026010 -id:A026010 - cp's OEIS frontend

A325698 Numbers with as many even as odd prime indices, counted with multiplicity.

1, 6, 14, 15, 26, 33, 35, 36, 38, 51, 58, 65, 69, 74, 77, 84, 86, 90, 93, 95, 106, 119, 122, 123, 141, 142, 143, 145, 156, 158, 161, 177, 178, 185, 196, 198, 201, 202, 209, 210, 214, 215, 216, 217, 219, 221, 225, 226, 228, 249, 262, 265, 278, 287, 291, 299

Offset: 1

Gus Wiseman, May 17 2019

nonn

These are Heinz numbers of the integer partitions counted by A045931.
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 integers in the multiplicative subgroup of positive rational numbers generated by the products of two consecutive primes (A006094). The sequence is closed under multiplication, prime shift (A003961), and - where the result is an integer - under division. Using these closures, all the terms can be derived from the presence of 6. For example, A003961(6) = 15, A003961(15) = 35, 6 * 35 = 210, 210/15 = 14. Closed also under A297845, since A297845 can be defined using squaring, prime shift and multiplication. - Peter Munn, Oct 05 2020

			The sequence of terms together with their prime indices begins:
    1: {}
    6: {1,2}
   14: {1,4}
   15: {2,3}
   26: {1,6}
   33: {2,5}
   35: {3,4}
   36: {1,1,2,2}
   38: {1,8}
   51: {2,7}
   58: {1,10}
   65: {3,6}
   69: {2,9}
   74: {1,12}
   77: {4,5}
   84: {1,1,2,4}
   86: {1,14}
   90: {1,2,2,3}
   93: {2,11}
   95: {3,8}

David A. Corneth, Table of n, a(n) for n = 1..10000

Positions of 0's in A195017.
A257992(n) = A257991(n).
Cf. A000712, A001222, A001405, A006094, A026010, A045931, A063886, A097613, A112798, A130780, A171966, A239241, A241638, A325700.
Closed under: A003961, A003991, A297845.
Subsequence of A028260, A332820.

Select[Range[100],Total[Cases[If[#==1,{},FactorInteger[#]],{p_,k_}:>k*(-1)^PrimePi[p]]]==0&]

is(n) = {my(v = vector(2), f = factor(n));for(i = 1, #f~,v[1 + primepi(f[i, 1])%2]+=f[i, 2]);v[1] == v[2]} \\ David A. Corneth, Oct 06 2020

from sympy import factorint, primepi
def ok(n):
    v = [0, 0]
    for p, e in factorint(n).items(): v[primepi(p)%2] += e
    return v[0] == v[1]
print([k for k in range(300) if ok(k)]) # Michael S. Branicky, Apr 16 2022 after David A. Corneth

A063886 Number of n-step walks on a line starting from the origin but not returning to it.

Original entry on oeis.org

1, 2, 2, 4, 6, 12, 20, 40, 70, 140, 252, 504, 924, 1848, 3432, 6864, 12870, 25740, 48620, 97240, 184756, 369512, 705432, 1410864, 2704156, 5408312, 10400600, 20801200, 40116600, 80233200, 155117520, 310235040, 601080390, 1202160780, 2333606220, 4667212440

Offset: 0

Henry Bottomley, Aug 28 2001

A Chebyshev transform of A007877(n+1). The g.f. is transformed to (1+x)/((1-x)(1+x^2)) under the mapping G(x)->(1/(1+x^2))G(1/(1+x^2)). - Paul Barry, Oct 12 2004
a(n-1) = 2*C(n-2, floor((n-2)/2)) is also the number of bit strings of length n in which the number of 00 substrings is equal to the number of 11 substrings. For example, when n = 4 we have 4 such bit strings: 0011, 0101, 1010, and 1100. - Angel Plaza, Apr 23 2009
Hankel transform is A120617. - Paul Barry, Aug 10 2009
The Hankel transform of a(n) is (-2)^C(n+1,2). The Hankel transform of (-1)^C(n+1,2)*a(n) is (-1)^C(n+1,2)*A164584(n). - Paul Barry, Aug 17 2009
For n > 1, a(n) is also the number of n-step walks starting from the origin and returning to it exactly once. - Geoffrey Critzer, Jan 24 2010
-a(n) is the Z-sequence for the Riordan array A130777. (See the W. Lang link under A006232 for A- and Z-sequences for Riordan matrices). - Wolfdieter Lang, Jul 12 2011
Number of subsets of {1,...,n} in which the even elements appear as often at even positions as at odd positions. - Gus Wiseman, Mar 17 2018

			a(4) = 6 because there are six length four walks that do not return to the origin: {-1, -2, -3, -4}, {-1, -2, -3, -2}, {-1, -2, -1, -2}, {1, 2, 1, 2}, {1, 2, 3, 2}, {1, 2, 3, 4}. There are also six such walks that return exactly one time: {-1, -2, -1, 0}, {-1, 0, -1, -2}, {-1, 0, 1, 2}, {1, 0, -1, -2}, {1, 0, 1, 2}, {1, 2, 1, 0}. - _Geoffrey Critzer_, Jan 24 2010
The a(5) = 12 subsets in which the even elements appear as often at even positions as at odd positions: {}, {1}, {3}, {5}, {1,3}, {1,5}, {2,4}, {3,5}, {1,2,4}, {1,3,5}, {2,4,5}, {1,2,4,5}. - _Gus Wiseman_, Mar 17 2018

Alois P. Heinz, Table of n, a(n) for n = 0..1000
Paul Barry, A Note on Riordan Arrays with Catalan Halves, arXiv:1912.01124 [math.CO], 2019.
Emeric Deutsch, Problem 11424, The American Mathematical Monthly, Vol. 116, No. 3 (March 2009), p. 277.
D. Perrin, A conjecture on rational sequences, pp. 267-274 of R. M. Capocelli, ed., Sequences, Springer-Verlag, NY 1990.

Apart from initial terms, same as A182027.
Cf. A000108, A000246, A000712, A000984, A001405, A001700, A002420, A006232, A007877, A026010, A028329, A045931, A047073, A089849, A097613, A106180, A120617, A130777, A130780, A138364, A164584, A171966, A239241, A300787, A300788, A300789.
Cf. A307768 (complementary event).

[1] cat [2*Binomial(n-1, Floor((n-1)/2)): n in [1..40]]; // G. C. Greubel, Jun 07 2023

seq(seq(binomial(2*j,j)*i, i=1..2),j=0..16); # Zerinvary Lajos, Apr 28 2007
# second Maple program:
a:= proc(n) option remember; `if`(n<2, n+1,
       4*a(n-2) +2*(a(n-1) -4*a(n-2))/n)
    end:
seq(a(n), n=0..40);  # Alois P. Heinz, Feb 10 2014
# third program:
A063886 := series(BesselI(0, 2*x)*(1 + x*2 + x*Pi*StruveL(1, 2*x)) - Pi*x*BesselI(1, 2*x)*StruveL(0, 2*x), x = 0, 34): seq(n!*coeff(A063886, x, n), n = 0 .. 33); # Mélika Tebni, Jun 17 2024

Table[Length[Select[Map[Accumulate, Strings[{-1, 1}, n]], Count[ #, 0] == 0 &]], {n, 0, 20}] (* Geoffrey Critzer, Jan 24 2010 *)
CoefficientList[Series[Sqrt[(1+2x)/(1-2x)],{x,0,40}],x] (* Harvey P. Dale, Apr 28 2016 *)

PARI
```
a(n)=(n==0)+2*binomial(n-1,(n-1)\2)
```

a(n) = 2^n*prod(k=0,n-1,(k/n+1/n)^((-1)^k)); \\ Michel Marcus, Dec 03 2013

from math import ceil
from sympy import binomial
def a(n):
    if n==0: return 1
    return 2*binomial(n-1,(n-1)//2)
print([a(n) for n in range(18)])
# David Nacin, Feb 29 2012

[2*binomial(n-1, (n-1)//2) + int(n==0) for n in range(41)] # G. C. Greubel, Jun 07 2023

G.f.: sqrt((1+2*x)/(1-2*x)).
a(n+1) = 2*C(n, floor(n/2)) = 2*A001405(n); a(2n) = C(2n, n) = A000984(n) = 4*a(2n-2)-|A002420(n)| = 4*a(2n-2)-2*A000108(n-1) = 2*A001700(n-1); a(2n+1) = 2*a(2n) = A028329(n).
2*a(n) = A047073(n+1).
a(n) = Sum_{k=0..n} abs(A106180(n,k)). - Philippe Deléham, Oct 06 2006
a(n) = Sum_{k=0..n} (k+1)binomial(n, (n-k)/2) ( 1-cos((k+1)*Pi/2) (1+(-1)^(n-k))/(n+k+2) ). - Paul Barry, Oct 12 2004
G.f.: 1/(1-2*x/(1+x/(1+x/(1-x/(1-x/(1+x/(1+x/(1-x/(1-x/(1+ ... (continued fraction). - Paul Barry, Aug 10 2009
G.f.: 1 + 2*x/(G(0)-x+x^2) where G(k)= 1 - 2*x^2 - x^4/G(k+1); (continued fraction, 1-step). - Sergei N. Gladkovskii, Aug 10 2012
D-finite with recurrence: n*a(n) = 2*a(n-1) + 4*(n-2)*a(n-2). - R. J. Mathar, Dec 03 2012
From Sergei N. Gladkovskii, Jul 26 2013: (Start)
G.f.: 1/G(0), where G(k) = 1 - 2*x/(1 + 2*x/(1 + 1/G(k+1) )); (continued fraction).
G.f.: G(0), where G(k) = 1 + 2*x/(1 - 2*x/(1 + 1/G(k+1) )); (continued fraction).
G.f.: W(0)/2*(1+2*x), where W(k) = 1 + 1/(1 - 2*x/(2*x + (k+1)/(x*(2*k+1))/W(k+1) )), abs(x) < 1/2; (continued fraction). (End)
a(n) = 2^n*Product_{k=0..n-1} (k/n + 1/n)^((-1)^k). - Peter Luschny, Dec 02 2013
G.f.: G(0), where G(k) = 1 + 2*x*(4*k+1)/((2*k+1)*(1+2*x) - (2*k+1)*(4*k+3)*x*(1+2*x)/((4*k+3)*x + (k+1)*(1+2*x)/G(k+1) )); (continued fraction). - Sergei N. Gladkovskii, Jan 19 2014
From Peter Bala, Mar 29 2024: (Start)
a(n) = 2^n * Sum_{k = 0..n} (-1)^(n+k)*binomial(1/2, k)*binomial(- 1/2, n-k) = 2^n * A000246(n)/n!.
a(n) = (1/2^n) * binomial(2*n, n) * hypergeom([-1/2, -n], [1/2 - n], -1). (End)
E.g.f.: BesselI(0, 2*x)*(1 + x*(2 + Pi)*StruveL(1, 2*x)) - Pi*x*BesselI(1, 2*x)*StruveL(0, 2*x). - Stefano Spezia, May 11 2024
a(n) = A089849(n) + A138364(n). - Mélika Tebni, Jun 17 2024
From Amiram Eldar, Aug 15 2025: (Start)
Sum_{n>=0} 1/a(n) = Pi/(3*sqrt(3)) + 2.
Sum_{n>=0} (-1)^n/a(n) = 2/3 + Pi/(9*sqrt(3)). (End)

A325700 Numbers with as many distinct even as distinct odd prime indices.

Original entry on oeis.org

1, 6, 12, 14, 15, 18, 24, 26, 28, 33, 35, 36, 38, 45, 48, 51, 52, 54, 56, 58, 65, 69, 72, 74, 75, 76, 77, 86, 93, 95, 96, 98, 99, 104, 106, 108, 112, 116, 119, 122, 123, 135, 141, 142, 143, 144, 145, 148, 152, 153, 158, 161, 162, 172, 175, 177, 178, 185, 192

Offset: 1

Gus Wiseman, May 17 2019

nonn

These are the Heinz numbers of the integer partitions counted by A241638.

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 sequence of terms together with their prime indices begins:
    1: {}
    6: {1,2}
   12: {1,1,2}
   14: {1,4}
   15: {2,3}
   18: {1,2,2}
   24: {1,1,1,2}
   26: {1,6}
   28: {1,1,4}
   33: {2,5}
   35: {3,4}
   36: {1,1,2,2}
   38: {1,8}
   45: {2,2,3}
   48: {1,1,1,1,2}
   51: {2,7}
   52: {1,1,6}
   54: {1,2,2,2}
   56: {1,1,1,4}
   58: {1,10}

A324967(n) = A324966(n).

Cf. A001221, A026010, A028260, A045931, A063886, A097613, A112798, A130780, A239241, A241638, A325698, A325699.

Select[Range[100],0==Total[(-1)^PrimePi/@First/@If[#==1,{},FactorInteger[#]]]&]

A097613 a(n) = binomial(2n-3,n-1) + binomial(2n-2,n-2).

Original entry on oeis.org

1, 2, 7, 25, 91, 336, 1254, 4719, 17875, 68068, 260338, 999362, 3848222, 14858000, 57500460, 222981435, 866262915, 3370764540, 13135064250, 51250632510, 200205672810, 782920544640, 3064665881940, 12007086477750, 47081501377326, 184753963255176, 725510446350004

Offset: 1

David Callan, Sep 20 2004

nonn

a(n) is the number of Dyck (2n-1)-paths with maximum pyramid size = n. A pyramid in a Dyck path is a maximal subpath of the form k upsteps immediately followed by k downsteps and its size is k.
a(n) is the total number of runs of peaks in all Dyck (n+1)-paths. A run of peaks is a maximal subpath of the form (UD)^k with k>=1. For example, a(2)=7 because the 5 Dyck 3-paths contain a total of 7 runs of peaks (in uppercase type): uuUDdd, uUDUDd, uUDdUD, UDuUDd, UDUDUD. - David Callan, Jun 07 2006
Binomial transform of A113682. - Paul Barry, Aug 21 2007
If Y is a fixed 2-subset of a (2n+1)-set X then a(n+1) is the number of n-subsets of X intersecting Y. - Milan Janjic, Oct 21 2007
Equals the Catalan sequence, A000108, convolved with A051924 prefaced with a 1: (1, 1, 4, 14, 50, ...). - Gary W. Adamson, May 15 2009
Central terms of triangle A209561. - Reinhard Zumkeller, Dec 26 2012
Also the number of compositions of 2*(n-1) in which the odd parts appear as many times in odd as in even positions. - Alois P. Heinz, May 26 2018

			a(2) = 2 because UUDDUD and UDUUDD each have maximum pyramid size = 2.

Michael De Vlieger, Table of n, a(n) for n = 1..1664
Paul Barry, Extensions of Riordan Arrays and Their Applications, Mathematics (2025) Vol. 13, No. 2, 242. See p. 16.
Gi-Sang Cheon, Hana Kim, and Louis W. Shapiro, Mutation effects in ordered trees, arXiv preprint arXiv:1410.1249 [math.CO], 2014.
Milan Janjic, Two Enumerative Functions
Toufik Mansour and I. L. Ramirez, Enumerations of polyominoes determined by Fuss-Catalan words, Australas. J. Combin. 81 (3) (2021) 447-457, table 2.
Lin Yang and Shengliang Yang, Protected Branches in Ordered Trees, J. Math. Study (2023) Vol. 56, No. 1, 1-17.

Same as A024482 except for first term.

Cf. A000108, A000712, A026010, A051924, A113682, A209561.

Flat(List([1..30], n->Binomial(2*n-3, n-1)+Binomial(2*n-2, n-2))); # Stefano Spezia, Oct 27 2018

a097613 n = a209561 (2 * n - 1) n  -- Reinhard Zumkeller, Dec 26 2012

[((3*n-2)*Catalan(n-1)+0^(n-1))/2: n in [1..40]]; // G. C. Greubel, Apr 04 2024

Z:=(1-z-sqrt(1-4*z))/sqrt(1-4*z)/2: Zser:=series(Z, z=0, 32): seq (ceil(coeff(Zser, z, n)), n=1..22); # Zerinvary Lajos, Jan 16 2007
a := n -> `if`(n=1, 1, (2-3*n)/(4-8*n)*binomial(2*n, n)):
seq(a(n), n=1..27); # Peter Luschny, Sep 06 2014

a[1]=1; a[n_] := (3n-2)(2n-3)!/(n!(n-2)!); Array[a, 27] (* Jean-François Alcover, Oct 27 2018 *)

a(n)=binomial(2*n-3,n-1)+binomial(2*n-2,n-2) \\ Charles R Greathouse IV, Aug 05 2013

@CachedFunction
def A097613(n):
    if n < 3: return n
    return (6*n-4)*(2*n-3)*A097613(n-1)/(n*(3*n-5))
[A097613(n) for n in (1..27)] # Peter Luschny, Sep 06 2014

G.f.: (x-1)*(1 - 1/sqrt(1-4*x))/2.
a(n) = ceiling(A051924(n)/2). - Zerinvary Lajos, Jan 16 2007
Integral representation as n-th moment of a signed weight function W(x) = W_a(x) + W_c(x), where W_a(x) = Dirac(x)/2 is the discrete (atomic) part, and  W_c(x) = (1/(2*Pi))*((x-1))*sqrt(1/(x*(4-x))) is the continuous part of W(x) defined on (0,4): a(n) = Integral_{x=-eps..eps} x^n*W_a(x) + Integral_{x=0..4} x^n*W_c(x) for any eps > 0, n >= 0. W_c(0) = -infinity, W_c(1) = 0 and W_c(4) = infinity. For 0 < x < 1, W_c(x) < 0, and for 1 < x < 4, W_c(x) > 0. - Karol A. Penson, Aug 05 2013
From Peter Luschny, Sep 06 2014: (Start)
a(n) = ((2-3*n)/(4-8*n))*binomial(2*n,n) for n >= 2.
D-finite with recurrence: a(n) = (6*n-4)*(2*n-3)*a(n-1)/(n*(3*n-5)) for n >= 3. (End)
a(n) ~ 3*2^(2*n-3)/sqrt(n*Pi). - Stefano Spezia, May 09 2023
From G. C. Greubel, Apr 04 2024: (Start)
a(n) = (1/2)*( (3*n-2)*A000108(n-1) + [n=1]).
E.g.f.: (1/2)*(-1+x + exp(2*x)*((1-x)*BesselI(0,2*x) + x*BesselI(1,2*x) )). (End)

A274230 Number of holes in a sheet of paper when you fold it n times and cut off the four corners.

Original entry on oeis.org

0, 0, 1, 3, 9, 21, 49, 105, 225, 465, 961, 1953, 3969, 8001, 16129, 32385, 65025, 130305, 261121, 522753, 1046529, 2094081, 4190209, 8382465, 16769025, 33542145, 67092481, 134193153, 268402689, 536821761, 1073676289, 2147385345

Offset: 0

Philippe Gibone, Jun 15 2016

The folds are always made so the longer side becomes the shorter side.
We could have counted not only the holes but also all the notches: 4, 6, 9, 15, 25, 45, 81, 153, 289, ... which has the formula a(n) = (2^ceiling(n/2) + 1) * (2^floor(n/2) + 1) and appears to match the sequence A183978. - Philippe Gibone, Jul 06 2016
The same sequence (0,0,1,3,9,21,49,...) turns up when you start with an isosceles right triangular piece of paper and repeatedly fold it in half, snipping corners as you go. Is there an easy way to see why the two questions have the same answer? - James Propp, Jul 05 2016
Reply from Tom Karzes, Jul 05 2016: (Start)
This case seems a little more complicated than the rectangular case, since with the triangle you alternate between horizontal/vertical folds vs. diagonal folds, and the resulting fold pattern is more complex, but I think the basic argument is essentially the same.
Note that with the triangle, the first hole doesn't appear until after you've made 3 folds, so if you start counting at zero folds, you have three leading zeros in the sequence: 0,0,0,1,3,9,21,... (End)
Also the number of subsets of {1,2,...,n} that contain both even and odd numbers. For example, a(3)=3 and the 3 subsets are {1,2}, {2,3}, {1,2,3}; a(4)=9 and the 9 subsets are {1,2}, {1,4}, {2,3}, {3,4}, {1,2,3}, {1,2,4}, {1,3,4}, {2,3,4}, {1,2,3,4}. (See comments in A052551 for the number of subsets of {1,2,...,n} that contain only odd and even numbers.) - Enrique Navarrete, Mar 26 2018
Also the number of integer compositions of n + 1 with an odd part other than the first or last. The complementary compositions are counted by A052955(n>0) = A027383(n) + 1. - Gus Wiseman, Feb 05 2022
Also the number of unit squares in the (n+1)-st iteration in the version of the dragon curve where the rotation directions alternate, so that any clockwise rotation is followed by a counterclockwise rotation, and vice versa (see image link below). - Talmon Silver, May 09 2023

Paolo P. Lava, Table of n, a(n) for n = 0..1000
Philippe Gibone, Illustration of a(0)-a(4) (idealized).
Talmon Silver, Illustration of alternating dragon.
N. J. A. Sloane, Illustration for a(4) = 9 (scan of an actual cut-up piece of paper)
N. J. A. Sloane, Illustration for a(5) = 21 (scan of an actual cut-up piece of paper)
Index entries for linear recurrences with constant coefficients, signature (3,0,-6,4).

Cf. A000225, A079667, A183978.
See A274626, A274627 for the three- and higher-dimensional analogs.
This is the main diagonal of A274635.
Counting fold lines instead of holes gives A027383.
Bisections are A060867 (even) and A134057 (odd).
Cf. A000712, A001405, A008549, A011782, A026010, A052955, A097613, A126869, A138364, A232580, A351003.

[(2^Ceiling(n/2)-1)*(2^Floor(n/2)-1 ): n in [0..35]]; // Vincenzo Librandi, Jul 02 2016

A274230:=n->(1+2^n-2^((n-3)/2)*(3-3*(-1)^n+2*sqrt(2)+2*(-1)^n*sqrt(2))): seq(A274230(n), n=0..50); # Wesley Ivan Hurt, Jul 07 2016

Table[(2^Ceiling@ # - 1) (2^Floor@ # - 1) &[n/2], {n, 0, 31}] (* Michael De Vlieger, Jun 30 2016 *)

a(n)=2^n+1-(n%2+2)<<(n\2) \\ Charles R Greathouse IV, Jul 05 2016

u(0) = 0; v(0) = 0; u(n+1) = v(n); v(n+1) = 2u(n) + 1; a(n) = u(n)*v(n).
a(n) = (2^ceiling(n/2) - 1)*(2^floor(n/2) - 1).
Proof from Tom Karzes, Jul 05 2016: (Start)
Let r be the number of times you fold along one axis and s be the number of times you fold along the other axis. So r is ceiling(n/2) and s is floor(n/2), where n is the total number of folds.
When unfolded, the resulting paper has been divided into a grid of (2^r) by (2^s) rectangles. The interior grid lines will have diamond-shaped holes where they intersect (assuming diagonal cuts).
There are (2^r-1) internal grid lines along one axis and (2^s-1) along the other. The total number of internal grid line intersections is therefore (2^r-1)*(2^s-1), or (2^ceiling(n/2)-1)*(2^floor(n/2)-1) as claimed. (End)
From Colin Barker, Jun 22 2016, revised by N. J. A. Sloane, Jul 05 2016: (Start)
It follows that:
a(n) = (2^(n/2)-1)^2 for n even, a(n) = 2^n+1-3*2^((n-1)/2) for n odd.
a(n) = 3*a(n-1)-6*a(n-3)+4*a(n-4) for n>3.
G.f.: x^2 / ((1-x)*(1-2*x)*(1-2*x^2)).
a(n) = (1+2^n-2^((n-3)/2)*(3-3*(-1)^n+2*sqrt(2)+2*(-1)^n*sqrt(2))). (End)
a(n) = A000225(n) - 2*A052955(n-2) for n > 1. - Yuchun Ji, Nov 19 2018
a(n) = A079667(2^(n-1)) for n >= 1. - J. M. Bergot, Jan 18 2021
a(n) = 2^(n-1) - A052955(n) = 2^(n-1) - A027383(n) - 1. - Gus Wiseman, Jan 29 2022
E.g.f.: cosh(x) + cosh(2*x) - 2*cosh(sqrt(2)*x) + sinh(x) + sinh(2*x) - 3*sinh(sqrt(2)*x)/sqrt(2). - Stefano Spezia, Apr 06 2022

A026009 Triangular array T read by rows: T(n,0) = 1 for n >= 0; T(1,1) = 1; and for n >= 2, T(n,k) = T(n-1,k-1) + T(n-1,k) for k = 1,2,...,[(n+1)/2]; T(n,n/2 + 1) = T(n-1,n/2) if n is even.

Original entry on oeis.org

1, 1, 1, 1, 2, 1, 1, 3, 3, 1, 4, 6, 3, 1, 5, 10, 9, 1, 6, 15, 19, 9, 1, 7, 21, 34, 28, 1, 8, 28, 55, 62, 28, 1, 9, 36, 83, 117, 90, 1, 10, 45, 119, 200, 207, 90, 1, 11, 55, 164, 319, 407, 297, 1, 12, 66, 219, 483, 726, 704, 297, 1, 13, 78, 285, 702, 1209, 1430, 1001, 1, 14, 91, 363, 987, 1911, 2639, 2431, 1001

Offset: 0

Clark Kimberling

			From _Jonathon Kirkpatrick_, Jul 01 2016: (Start)
Triangle begins:
  1;
  1,  1;
  1,  2,  1;
  1,  3,  3;
  1,  4,  6,   3;
  1,  5, 10,   9;
  1,  6, 15,  19,   9;
  1,  7, 21,  34,  28;
  1,  8, 28,  55,  62,   28;
  1,  9, 36,  83, 117,   90;
  1, 10, 45, 119, 200,  207,   90;
  1, 11, 55, 164, 319,  407,  297;
  1, 12, 66, 219, 483,  726,  704,  297;
  1, 13, 78, 285, 702, 1209, 1430, 1001;
  ... (End)

G. C. Greubel, Rows n = 0..100 of the triangle, flattened

Diagonals of this sequence: A000217, A000245, A026012, A026013, A026014, A026015, A026016, A026017, A026018, A026019, A026020, A026021.

Sums involving this sequence: A026010, A027287, A027288, A027289, A027290, A027291, A027292.

[1] cat [Binomial(n,k) - Binomial(n,k-3): k in [0..Floor((n+2)/2)], n in [1..15]]; // G. C. Greubel, Mar 18 2021

T[n_, k_]:= Binomial[n, k] - Binomial[n, k-3];
Join[{1}, Table[T[n, k], {n,14}, {k,0,Floor[(n+2)/2]}]//Flatten] (* G. C. Greubel, Mar 18 2021 *)

[1]+flatten([[binomial(n,k) - binomial(n,k-3) for k in (0..(n+2)//2)] for n in (1..15)]) # G. C. Greubel, Mar 18 2021

T(n, k) = binomial(n, k) - binomial(n, k-3). - Darko Marinov (marinov(AT)lcs.mit.edu), May 17 2001

Sum_{k=0..floor((n+2)/2)} T(n, k) = A026010(n). - G. C. Greubel, Mar 18 2021

A005774 Number of directed animals of size n (k=1 column of A038622); 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, where s(0) = 2; also sum of row n+1 of array T in A026323.

Original entry on oeis.org

0, 1, 3, 9, 26, 75, 216, 623, 1800, 5211, 15115, 43923, 127854, 372749, 1088283, 3181545, 9312312, 27287091, 80038449, 234988827, 690513030, 2030695569, 5976418602, 17601021837, 51869858544, 152951628725, 451271872701, 1332147482253

Offset: 0

Simon Plouffe

Number of ordered trees with n+1 edges, having root degree at least 2 and nonroot outdegrees at most 2. - Emeric Deutsch, Aug 02 2002
From Petkovsek's algorithm, this recurrence does not have any closed form solutions. So there is no hypergeometric closed form for a(n). - Herbert S. Wilf
Sum of two consecutive trinomial coefficients starting two positions before central one. Example: a(4) = 10+16 and (1 + x + x^2)^4 = ... + 10*x^2 + 16*x^3 + 19*x^4 + ... - David Callan, Feb 07 2004
Image of n (A001477) under the Motzkin related matrix A107131. Binomial transform of A037952. - Paul Barry, May 12 2005
a(n) = total number of ascents (maximal runs of consecutive upsteps) in all Motzkin (n+1)-paths. For example, the 9 Motzkin 4-paths are FFFF, FFUD, FUDF, FUFD, UDFF, UDUD, UFDF, UFFD, UUDD and they contain a total of 9 ascents and so a(3)=9 (U=upstep, D=downstep, F=flatstep). - David Callan, Aug 16 2006
Image of the sequence (0,1,2,3,3,3,...) under the array A122896. - Paul Barry, Sep 18 2006
This is some kind of Motzkin transform of A079978 because the substitution x-> x*A001006(x) in the independent variable of the g.f. A079978(x) yields 1,0 followed by this sequence here. - R. J. Mathar, Nov 08 2008

			G.f.: x + 3*x^2 + 9*x^3 + 26*x^4 + 75*x^5 + 216*x^6 + 623*x^7 + ...

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

Reinhard Zumkeller, Table of n, a(n) for n = 0..1000
P. Barry, Riordan arrays, generalized Narayana triangles, and series reversion, Linear Algebra and its Applications, 491 (2016) 343-385.
D. Gouyou-Beauchamps, G. Viennot, Equivalence of the two-dimensional directed animal problem to a one-dimensional path problem, Adv. in Appl. Math. 9 (1988), no. 3, 334-357.
Christian Krattenthaler, Daniel Yaqubi, Some determinants of path generating functions, II, arXiv:1802.05990 [math.CO], 2018; Adv. Appl. Math. 101 (2018), 232-265.
Simon Plouffe, Approximations de Séries Génératrices et Quelques Conjectures, Dissertation, Université du Québec à Montréal, 1992.
Simon Plouffe, Une méthode pour obtenir la fonction génératrice d'une série, FPSAC 1993, Florence. Formal Power Series and Algebraic Combinatorics.

Cf. A038622, A098494, A026010, A005773, A005775, A066822.

a005774 0 = 0
a005774 n = a038622 n 1 -- Reinhard Zumkeller, Feb 26 2013

seq( add(binomial(i,k+1)*binomial(i-k,k), k=0..floor(i/2)), i=0..30 ); # Detlef Pauly (dettodet(AT)yahoo.de), Nov 09 2001
seq(simplify(GegenbauerC(n-2,-n,-1/2) + GegenbauerC(n-1,-n,-1/2)), n=0..27); # Peter Luschny, May 12 2016

CoefficientList[Series[(1-x-Sqrt[1-2x-3x^2])/(x(1-3x+Sqrt[1-2x-3x^2])),{x,0,30}],x] (* Harvey P. Dale, Sep 20 2011 *)
RecurrenceTable[{a[0]==0, a[1]==1,a[n]==(2n(n+1)a[n-1]+3n(n-1)a[n-2])/ ((n+2)(n-1))},a,{n,30}] (* Harvey P. Dale, Nov 09 2012 *)

s=[0,1]; {A005774(n)=k=(2*(n+2)*(n+1)*s[2]+3*(n+1)*n*s[1])/((n+3)*n); s[1]=s[2]; s[2]=k; k}

{a(n) = if( n<2, n>0, (2 * (n+1) * n *a(n-1) + 3 * (n-1) * n * a(n-2)) / (n+2) / (n-1))}; /* Michael Somos, May 01 2003 */

Inverse binomial transform of [ 0, 1, 5, 21, 84, ... ] (A002054). - John W. Layman
D-finite with recurrence (n+2)*(n-1)*a(n) = 2*n*(n+1)*a(n-1) + 3*n*(n-1)*a(n-2) for all n in Z. - Michael Somos, May 01 2003
E.g.f.: exp(x)*(BesselI(1, 2*x)+BesselI(2, 2*x)). - Vladeta Jovovic, Jan 01 2004
G.f.: (1-x-sqrt(1-2x-3x^2))/(x(1-3x+sqrt(1-2x-3x^2))); a(n)= Sum_{k=0..n} C(k+1, n-k+1)*C(n, k)*k/(k+1); a(n) = Sum_{k=0..n} C(n, k)*C(k, floor((k-1)/2)). - Paul Barry, May 12 2005
Starting (1, 3, 9, 26, ...) = binomial transform of A026010: (1, 2, 4, 7, 14, 25, 50, 91, ...). - Gary W. Adamson, Oct 22 2007
a(n)*(2+n) = (4+4*n)*a(n-1) - n*a(n-2) + (12-6*n)*a(n-3). - Simon Plouffe, Feb 09 2012
a(n) ~ 3^(n+1/2) / sqrt(Pi*n). - Vaclav Kotesovec, Mar 10 2014
0 = a(n)*(+36*a(n+1) + 18*a(n+2) - 96*a(n+3) + 30*a(n+4)) + a(n+1)*(-6*a(n+1) + 49*a(n+2) - 26*a(n+3) + 3*a(n+4)) + a(n+2)*(+15*a(n+3) - 8*a(n+4)) + a(n+3)*(a(n+4)) if n >= 0. - Michael Somos, Aug 06 2014
a(n) = GegenbauerC(n-2,-n,-1/2) + GegenbauerC(n-1,-n,-1/2). - Peter Luschny, May 12 2016

Further descriptions from Clark Kimberling

A300788 Number of strict integer partitions of n in which the even parts appear as often at even positions as at odd positions.

Original entry on oeis.org

1, 1, 0, 1, 1, 1, 2, 2, 3, 3, 4, 5, 6, 7, 8, 10, 12, 14, 16, 19, 23, 26, 30, 35, 42, 47, 54, 62, 73, 82, 94, 107, 124, 139, 158, 179, 206, 230, 260, 293, 334, 372, 420, 470, 532, 591, 664, 740, 835, 924, 1034, 1148, 1288, 1422, 1588, 1756, 1962, 2161, 2404

Offset: 0

Gus Wiseman, Mar 12 2018

nonn

			The a(9) = 3 strict partitions: (9), (621), (531). Missing are: (81), (72), (63), (54), (432).

Alois P. Heinz, Table of n, a(n) for n = 0..4000 (first 291 terms from Fausto A. C. Cariboni)

Cf. A000712, A000898, A001405, A026010, A045931, A063886, A097613, A130780, A171966, A239241, A300787, A300789.

cobal[y_]:=Sum[(-1)^x,{x,Join@@Position[y,_?EvenQ]}];
Table[Length[Select[IntegerPartitions[n],cobal[#]===0&&UnsameQ@@#&]],{n,0,40}]

a(41)-a(58) from Alois P. Heinz, Mar 13 2018

A240608 Number A(n,k) of n-length words w over a k-ary alphabet such that w is empty or a prefix z concatenated with letter a_i and i=1 or 0 < #(z,a_{i-1}) >= #(z,a_i), where #(z,a_i) counts the occurrences of the i-th letter in z; square array A(n,k), n>=0, k>=0, read by antidiagonals.

Original entry on oeis.org

1, 1, 0, 1, 1, 0, 1, 1, 1, 0, 1, 1, 2, 1, 0, 1, 1, 2, 4, 1, 0, 1, 1, 2, 5, 7, 1, 0, 1, 1, 2, 5, 13, 14, 1, 0, 1, 1, 2, 5, 14, 35, 25, 1, 0, 1, 1, 2, 5, 14, 45, 94, 50, 1, 0, 1, 1, 2, 5, 14, 46, 149, 254, 91, 1, 0, 1, 1, 2, 5, 14, 46, 164, 509, 688, 182, 1, 0, 1, 1, 2, 5, 14, 46, 165, 629, 1756, 1872, 336, 1, 0

Offset: 0

Alois P. Heinz, Apr 09 2014

			Square array A(n,k) begins:
  1, 1,  1,   1,    1,    1,    1,    1,    1, ...
  0, 1,  1,   1,    1,    1,    1,    1,    1, ...
  0, 1,  2,   2,    2,    2,    2,    2,    2, ...
  0, 1,  4,   5,    5,    5,    5,    5,    5, ...
  0, 1,  7,  13,   14,   14,   14,   14,   14, ...
  0, 1, 14,  35,   45,   46,   46,   46,   46, ...
  0, 1, 25,  94,  149,  164,  165,  165,  165, ...
  0, 1, 50, 254,  509,  629,  650,  651,  651, ...
  0, 1, 91, 688, 1756, 2511, 2742, 2770, 2771, ...

Alois P. Heinz, Antidiagonals n = 0..36, flattened

Columns k=0-10 give: A000007, A000012, A026010(n-1) for n>0, A240609, A240610, A240611, A240612, A240613, A240614, A240615, A240616.
Main diagonal gives A240617.
Cf. A182172.

b:= proc(n, k, l) option remember; `if`(n=0, 1, `if`(nops(l) b(n, min(k, n), []):
seq(seq(A(n, d-n), n=0..d), d=0..14);

b[n_, k_, l_List] := b[n, k, l] = If[n == 0, 1, If[Length[l] l[[i]]+1]], 0], {i, 1, Length[l]}]]; A[n_, k_] := b[n, Min[k, n], {}]; Table[ Table[A[n, d-n], {n, 0, d}], {d, 0, 14}] // Flatten (* Jean-François Alcover, Jan 19 2015, after Alois P. Heinz *)

A300787 Number of integer partitions of n in which the even parts appear as often at even positions as at odd positions.

Original entry on oeis.org

1, 1, 1, 2, 3, 4, 6, 8, 12, 15, 21, 27, 38, 47, 63, 79, 106, 130, 170, 209, 272, 330, 422, 512, 653, 784, 986, 1183, 1482, 1765, 2191, 2604, 3218, 3804, 4666, 5504, 6726, 7898, 9592, 11240, 13602, 15880, 19122, 22277, 26733, 31048, 37102, 43003, 51232, 59220

Offset: 0

Gus Wiseman, Mar 12 2018

nonn

			The a(7) = 8 partitions: (7), (511), (421), (331), (322), (31111), (22111), (1111111). Missing are: (61), (52), (43), (4111), (3211), (2221), (211111).

Even- and odd-indexed terms are A006330 and A001523 respectively, which add up to A000712.

Cf. A000898, A001405, A026010, A045931, A058696, A063886, A097613, A130780, A171966, A239241, A300788, A300789.

cobal[y_]:=Sum[(-1)^x,{x,Join@@Position[y,_?EvenQ]}];
Table[Length[Select[IntegerPartitions[n],cobal[#]===0&]],{n,0,50}]

cp's OEIS Frontend

A325698 Numbers with as many even as odd prime indices, counted with multiplicity.

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A063886 Number of n-step walks on a line starting from the origin but not returning to it.

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A325700 Numbers with as many distinct even as distinct odd prime indices.

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A097613 a(n) = binomial(2n-3,n-1) + binomial(2n-2,n-2).

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A274230 Number of holes in a sheet of paper when you fold it n times and cut off the four corners.

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A026009 Triangular array T read by rows: T(n,0) = 1 for n >= 0; T(1,1) = 1; and for n >= 2, T(n,k) = T(n-1,k-1) + T(n-1,k) for k = 1,2,...,[(n+1)/2]; T(n,n/2 + 1) = T(n-1,n/2) if n is even.

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