A026641 -id:A026641 - cp's OEIS frontend

A035317 Pascal-like triangle associated with A000670.

1, 1, 1, 1, 2, 2, 1, 3, 4, 2, 1, 4, 7, 6, 3, 1, 5, 11, 13, 9, 3, 1, 6, 16, 24, 22, 12, 4, 1, 7, 22, 40, 46, 34, 16, 4, 1, 8, 29, 62, 86, 80, 50, 20, 5, 1, 9, 37, 91, 148, 166, 130, 70, 25, 5, 1, 10, 46, 128, 239, 314, 296, 200, 95, 30, 6, 1, 11, 56, 174, 367, 553, 610, 496, 295, 125

Offset: 0

N. J. A. Sloane

From Johannes W. Meijer, Jul 20 2011: (Start)
The triangle sums, see A180662 for their definitions, link this "Races with Ties" triangle with several sequences, see the crossrefs. Observe that the Kn4 sums lead to the golden rectangle numbers A001654 and that the Fi1 and Fi2 sums lead to the Jacobsthal sequence A001045.
The series expansion of G(x, y) = 1/((y*x-1)*(y*x+1)*((y+1)*x-1)) as function of x leads to this sequence, see the second Maple program. (End)
T(2n,k) = the number of hatted frog arrangements with k frogs on the 2xn grid. See the linked paper "Frogs, hats and common subsequences". - Chris Cox, Apr 12 2024

			Triangle begins:
  1;
  1,  1;
  1,  2,  2;
  1,  3,  4,   2;
  1,  4,  7,   6,   3;
  1,  5, 11,  13,   9,   3;
  1,  6, 16,  24,  22,  12,   4;
  1,  7, 22,  40,  46,  34,  16,   4;
  1,  8, 29,  62,  86,  80,  50,  20,  5;
  1,  9, 37,  91, 148, 166, 130,  70, 25,  5;
  1, 10, 46, 128, 239, 314, 296, 200, 95, 30, 6;
  ...

Vincenzo Librandi, Rows n = 0..100, flattened
Joseph Briggs, Alex Parker, Coy Schwieder, and Chris Wells, Frogs, hats and common subsequences, arXiv preprint arXiv:2404.07285 [math.CO], 2024. See p. 28.
A. Hlavác, M. Marvan, Nonlocal conservation laws of the constant astigmatism equation, arXiv preprint arXiv:1602.06861 [nlin.SI], 2016.
E. Mendelson, Races with Ties, Math. Mag. 55 (1982), 170-175.
Index entries for triangles and arrays related to Pascal's triangle

Row sums are A000975, diagonal sums are A080239.
Central terms are A014300.
Similar to the triangles A059259, A080242, A108561, A112555.
Cf. A059260.
Triangle sums (see the comments): A000975 (Row1), A059841 (Row2), A080239 (Kn11), A052952 (Kn21), A129696 (Kn22), A001906 (Kn3), A001654 (Kn4), A001045 (Fi1, Fi2), A023435 (Ca2), Gi2 (A193146), A190525 (Ze2), A193147 (Ze3), A181532 (Ze4). - Johannes W. Meijer, Jul 20 2011
Cf. A181971.

a035317 n k = a035317_tabl !! n !! k
a035317_row n = a035317_tabl !! n
a035317_tabl = map snd $ iterate f (0, [1]) where
   f (i, row) = (1 - i, zipWith (+) ([0] ++ row) (row ++ [i]))
-- Reinhard Zumkeller, Jul 09 2012

A035317 := proc(n,k): add((-1)^(i+k) * binomial(i+n-k+1, i), i=0..k) end: seq(seq(A035317(n,k), k=0..n), n=0..10); # Johannes W. Meijer, Jul 20 2011
A035317 := proc(n,k): coeff(coeftayl(1/((y*x-1)*(y*x+1)*((y+1)*x-1)), x=0, n), y, k) end: seq(seq(A035317(n,k), k=0..n), n=0..10); # Johannes W. Meijer, Jul 20 2011

t[n_, k_] := (-1)^k*(((-1)^k*(n+2)!*Hypergeometric2F1[1, n+3, k+2, -1])/((k+1)!*(n-k+1)!) + 2^(k-n-2)); Flatten[ Table[ t[n, k], {n, 0, 11}, {k, 0, n}]] (* Jean-François Alcover, Dec 14 2011, after Johannes W. Meijer *)

{T(n,k)=if(n==k,(n+2)\2,if(k==0,1,if(n>k,T(n-1,k-1)+T(n-1,k))))}
for(n=0,12,for(k=0,n,print1(T(n,k),","));print("")) \\ Paul D. Hanna, Jul 18 2012

def A035317_row(n):
    @cached_function
    def prec(n, k):
        if k==n: return 1
        if k==0: return 0
        return -prec(n-1,k-1)-sum(prec(n,k+i-1) for i in (2..n-k+1))
    return [(-1)^k*prec(n+2, k) for k in (1..n)]
for n in (1..11): print(A035317_row(n)) # Peter Luschny, Mar 16 2016

T(n,k) = Sum_{j=0..floor(n/2)} binomial(n-2j, k-2j). - Paul Barry, Feb 11 2003
From Johannes W. Meijer, Jul 20 2011: (Start)
T(n, k) = Sum_{i=0..k}((-1)^(i+k) * binomial(i+n-k+1,i)). (Mendelson)
T(n, k) = T(n-1, k-1) + T(n-1, k) with T(n, 0) = 1 and T(n, n) = floor(n/2) + 1. (Mendelson)
Sum_{k = 0..n}((-1)^k * (n-k+1)^n * T(n, k)) = A000670(n). (Mendelson)
T(n, n-k) = A128176(n, k); T(n+k, n-k) = A158909(n, k); T(2*n-k, k) = A092879(n, k). (End)
T(2*n+1,n) = A014301(n+1); T(2*n+1,n+1) = A026641(n+1). - Reinhard Zumkeller, Jul 19 2012

More terms from James Sellers

A014301 Number of internal nodes of even outdegree in all ordered rooted trees with n edges.

Original entry on oeis.org

0, 1, 3, 11, 40, 148, 553, 2083, 7896, 30086, 115126, 442118, 1703052, 6577474, 25461493, 98759971, 383751472, 1493506534, 5820778858, 22714926826, 88745372992, 347087585824, 1358789148058, 5324148664846, 20878676356240, 81937643449468, 321786401450268

Offset: 1

Emeric Deutsch

nonn

Number of protected vertices in all ordered rooted trees with n edges. A protected vertex in an ordered tree is a vertex at least 2 edges away from its leaf descendants. - Emeric Deutsch, Aug 20 2008
1,3,11,... gives the diagonal sums of A111418. Hankel transform of a(n) is A128834. Hankel transform of a(n+1) is A187340. - Paul Barry, Mar 08 2011
a(n) = A035317(2*n-1,n-1) for n > 0. - Reinhard Zumkeller, Jul 19 2012
Apparently the number of peaks in all Dyck paths of semilength n+1 that are the same height as the preceding peak. - David Scambler, Apr 22 2013
Define an infinite triangle by T(n,0)=A001045(n) (the first column) and T(r,c) = Sum_{k=c-1..r} T(k,c-1) (the sum of all the terms in the preceding column down to row r). Then T(n,n)=a(n+1). The triangle is 0; 1,1; 1,2,3; 3,5,8,11; 5,10,18,29,40; 11,21,39,68,108,148;... Example: T(5,2)=39=the sum of the terms in column 1 from T(1,1) to T(5,1), namely, 1+2+5+10+21. - J. M. Bergot, May 17 2013
Also for n>=1 the number of unimodal functions f:[n]->[n] with f(1)<>1 and f(i)<>f(i+1). a(4) = 11: [2,3,2,1], [2,3,4,1], [2,3,4,2], [2,3,4,3], [2,4,2,1], [2,4,3,1], [2,4,3,2], [3,4,2,1], [3,4,3,1], [3,4,3,2], [4,3,2,1]. - Alois P. Heinz, May 23 2013

G. C. Greubel, Table of n, a(n) for n = 1..1000
Gi-Sang Cheon and Louis W. Shapiro, Protected points in ordered trees, Appl. Math. Letters, 21, 2008, 516-520.
Sergi Elizalde, Symmetric peaks and symmetric valleys in Dyck paths, arXiv:2008.05669 [math.CO], 2020, Corollary 3.4.
Torleiv Kløve, Spheres of Permutations under the Infinity Norm - Permutations with limited displacement, Reports in Informatics, Department of Informatics, University of Bergen, Norway, no. 376, November 2008.
Index entries for sequences related to rooted trees

Cf. A059481, A026641, A143362, A143363.

[(1/2)*(&+[(-1)^(n-k)*Binomial(n+k-1,k): k in [0..n]]): n in [1..30]]; // G. C. Greubel, Jan 15 2018

Rest[CoefficientList[Series[(1-2*x-Sqrt[1-4*x])/(3*Sqrt[1-4*x]-1+4*x), {x, 0, 50}], x]] (* G. C. Greubel, Jan 15 2018 *)

x='x+O('x^30); Vec((1-2*x-sqrt(1-4*x))/(3*sqrt(1-4*x)-1+4*x)) \\ G. C. Greubel, Jan 15 2018

from itertools import count, islice
def A014301_gen(): # generator of terms
    yield from (0,1)
    a, b, c = 0, 3, 1
    for n in count(1):
        yield ((b:=b*((n<<1)+3<<1)//(n+2))-(a:=(c:=c*((n<<2)+2)//(n+2))-a>>1))//3
A014301_list = list(islice(A014301_gen(),20)) # Chai Wah Wu, Apr 27 2023

a(n) = binomial(2*n-1, n)/3 - A000957(n)/3;
a(n) = (1/2)*Sum_{k=0..n} (-1)^(n-k)*binomial(n+k-1, k). - Vladeta Jovovic, Aug 28 2002
From Emeric Deutsch, Jan 26 2004: (Start)
G.f.: (1-2*z-sqrt(1-4*z))/(3*sqrt(1-4*z)-1+4*z).
a(n) = [A026641(n) - A026641(n-1)]/3 for n>1. (End)
a(n) = (1/2)*Sum_{j=0..floor(n/2)} binomial(2n-2j-2, n-2).
a(n) = Sum_{k=0..n} (-1)^(n-k)*C(n+k,k-1). - Paul Barry, Jul 18 2006
D-finite with recurrence: 2*n*a(n) +(-9*n+8)*a(n-1) +(3*n-16)*a(n-2) +2*(2*n-5)*a(n-3)=0. - R. J. Mathar, Dec 03 2012

A183160 a(n) = Sum_{k=0..n} C(n+k,n-k)C(2n-k,k).

Original entry on oeis.org

1, 2, 11, 62, 367, 2232, 13820, 86662, 548591, 3498146, 22436251, 144583496, 935394436, 6071718512, 39523955552, 257913792342, 1686627623151, 11050540084902, 72522925038257, 476669316338542, 3137209052543927, 20672732229560032, 136374124374593072, 900541325129687272

Offset: 0

Paul D. Hanna, Dec 27 2010

nonn

			G.f.: A(x) = 1 + 2*x + 11*x^2 + 62*x^3 + 367*x^4 + 2232*x^5 +...
A(x)^(1/2) = 1 + x + 5*x^2 + 26*x^3 + 145*x^4 + 841*x^5 + 5006*x^6 +...+ A183161(n)*x^n +...
Given triangle A085478(n,k) = C(n+k,n-k), which begins:
  1;
  1,  1;
  1,  3,  1;
  1,  6,  5,  1;
  1, 10, 15,  7, 1;
  1, 15, 35, 28, 9, 1; ...
ILLUSTRATE formula a(n) = Sum_{k=0..n} A085478(n,k)*A085478(n,n-k):
a(2) = 11 = 1*1 + 3*3 + 1*1;
a(3) = 62 = 1*1 + 6*5 + 5*6 + 1*1;
a(4) = 367 = 1*1 + 10*7 + 15*15 + 7*10 + 1*1;
a(5) = 2232 = 1*1 + 15*9 + 35*28 + 28*35 + 9*15 + 1*1; ...

Vincenzo Librandi, Table of n, a(n) for n = 0..200

Cf. A001764, A085478, A183161, A199033, A218622.
Cf. A026641, A147855, A371753.
Cf. A005809, A006256, A160906, A226751.
Cf. A171822.

[(&+[Binomial(n+k, 2*k)*Binomial(2*n-k, k): k in [0..n]]): n in [0..25]]; // G. C. Greubel, Feb 22 2021

Table[Sum[Binomial[n+k,n-k]Binomial[2n-k,k],{k,0,n}],{n,0,20}] (* Harvey P. Dale, Jul 19 2011 *)
Table[HypergeometricPFQ[{-n, -n, 1/2 -n, n+1}, {1/2, 1, -2*n}, 1], {n, 0, 25}] (* G. C. Greubel, Feb 22 2021 *)

{a(n)=sum(k=0,n,binomial(n+k,n-k)*binomial(2*n-k,k))}

{a(n)=local(G=1); for(i=0, n, G=1+x*G^3+O(x^(n+1))); polcoeff(1/(1-2*x*G^2-3*x^2*G^4), n)} \\ Paul D. Hanna, Nov 03 2012

{a(n)=local(G=1); for(i=0, n, G=1+x*G^3+O(x^(n+1))); polcoeff(1/(1+3*x*G-5*x*G^2), n)} \\ Paul D. Hanna, Jun 16 2013
for(n=0, 30, print1(a(n), ", "))

a = lambda n: binomial(3*n+1,n)*hypergeometric([1,-n],[2*n+2],2)
[simplify(a(n)) for n in range(26)] # Peter Luschny, May 19 2015

a(n) = Sum_{k=0..n} A085478(n,k)*A085478(n,n-k).
Self-convolution of A183161 (an integer sequence):
a(n) = Sum_{k=0..n} A183161(k)*A183161(n-k).
a(n) = Sum_{k=0..n} binomial(2*n+k,k) * cos((n+k)*Pi). - Arkadiusz Wesolowski, Apr 02 2012
Recurrence: 320*n*(2*n-1)*a(n) = 8*(346*n^2 + 79*n - 327)*a(n-1) + 6*(1688*n^2-6241*n+5981)*a(n-2) + 261*(3*n-7)*(3*n-5)*a(n-3). - Vaclav Kotesovec, Oct 20 2012
a(n) ~ 3^(3*n+3/2)/(2^(2*n+3)*sqrt(Pi*n)). - Vaclav Kotesovec, Oct 20 2012
...
G.f.: A(x) = 1/(1 - 2*x*G(x)^2 - 3*x^2*G(x)^4), where G(x) = 1 + x*G(x)^3 = g.f. of A001764. - Paul D. Hanna, Nov 03 2012
G.f.: A(x) = 1 + x*d/dx { log( G(x)^5/(1+x*G(x)^2) )/2 }, where G(x) = 1 + x*G(x)^3 = g.f. of A001764. - Paul D. Hanna, Nov 04 2012
G.f.: A(x) = 1/(1 + 3*x*G(x) - 5*x*G(x)^2), where G(x) = 1 + x*G(x)^3 = g.f. of A001764. - Paul D. Hanna, Jun 16 2013
a(n) = C(3*n+1,n)*Hyper2F1([1,-n],[2*n+2],2). - Peter Luschny, May 19 2015
a(n) = [x^n] 1/((1 - x^2)*(1 - x)^(2*n)). - Ilya Gutkovskiy, Oct 25 2017
From G. C. Greubel, Feb 22 2021: (Start)
a(n) = Sum_{k=0..n} A171822(n, k).
a(n) = Hypergeometric 4F3([-n, -n, 1/2 -n, n+1], [1/2, 1, -2*n], 1). (End)
a(n) = Sum_{k=0..floor(n/2)} binomial(3*n-2*k-1,n-2*k). - Seiichi Manyama, Apr 05 2024
a(n) = Sum_{k=0..n} (-2)^(n-k) * binomial(3*n+1,k). - Seiichi Manyama, Aug 03 2025
From Seiichi Manyama, Aug 14 2025: (Start)
a(n) = Sum_{k=0..n} (-1)^k * 2^(n-k) * binomial(3*n+1,k) * binomial(3*n-k,n-k).
G.f.: g^2/((-1+2*g) * (3-2*g)) where g = 1+x*g^3 is the g.f. of A001764. (End)
G.f.: B(x)^2/(1 + 4*(B(x)-1)/3), where B(x) is the g.f. of A005809. - Seiichi Manyama, Aug 15 2025

A026637 Triangular array T read by rows: T(n,0) = T(n,n) = 1 for n >= 0, T(n,1) = T(n,n-1) = floor((3*n-1)/2) for n >= 1, otherwise T(n,k) = T(n-1,k-1) + T(n-1,k) for 2 <= k <= n-2, n >= 4.

Original entry on oeis.org

1, 1, 1, 1, 2, 1, 1, 4, 4, 1, 1, 5, 8, 5, 1, 1, 7, 13, 13, 7, 1, 1, 8, 20, 26, 20, 8, 1, 1, 10, 28, 46, 46, 28, 10, 1, 1, 11, 38, 74, 92, 74, 38, 11, 1, 1, 13, 49, 112, 166, 166, 112, 49, 13, 1, 1, 14, 62, 161, 278, 332, 278, 161, 62, 14, 1, 1, 16, 76, 223, 439, 610, 610, 439, 223, 76, 16, 1

Offset: 0

Clark Kimberling

T(n, k) = number of paths from (0, 0) to (n-k, k) in directed graph having vertices (i, j) and edges (i, j)-to-(i+1, j) and (i, j)-to-(i, j+1) for i, j >= 0 and edges (i, j)-to-(i+1, j+1) for i=0, j >= 1 and odd and for j=0, i >= 1 and odd.

See A228053 for a sequence with many terms in common with this one. - T. D. Noe, Aug 07 2013

			Triangle begins as:
  1;
  1,  1;
  1,  2,  1;
  1,  4,  4,   1;
  1,  5,  8,   5,   1;
  1,  7, 13,  13,   7,   1;
  1,  8, 20,  26,  20,   8,   1;
  1, 10, 28,  46,  46,  28,  10,   1;
  1, 11, 38,  74,  92,  74,  38,  11,  1;
  1, 13, 49, 112, 166, 166, 112,  49, 13,  1;
  1, 14, 62, 161, 278, 332, 278, 161, 62, 14,  1;

Reinhard Zumkeller, Rows n = 0..100 of table, flattened

Cf. A026638, A026639, A026640, A026641, A026642, A026643, A026644.
Cf. A026966, A026967, A026968, A026969, A026970, A228053.
Sums include: A000007 (alternating sign row), A026644 (row), A026645, A026646, A026647 (diagonal).

a026637 n k = a026637_tabl !! n !! k
a026637_row n = a026637_tabl !! n
a026637_tabl = [1] : [1,1] : map (fst . snd)
   (iterate f (0, ([1,2,1], [0,1,1,0]))) where
   f (i, (xs, ws)) = (1 - i,
     if i == 1 then (ys, ws) else (zipWith (+) ys ws, ws'))
        where ys = zipWith (+) ([0] ++ xs) (xs ++ [0])
              ws' = [0,1,0,0] ++ drop 2 ws
-- Reinhard Zumkeller, Aug 08 2013

function T(n,k) // T = A026637
   if k eq 0 or k eq n then return 1;
   elif k eq 1 or k eq n-1 then return Floor((3*n-1)/2);
   else return T(n-1, k) + T(n-1, k-1);
   end if;
end function;
[T(n,k): k in [0..n], n in [0..15]]; // G. C. Greubel, Jun 28 2024

A026637 := proc(n,k)
      option remember;
      if k=0 or k=n then
        1
    elif k=1 or k=n-1 then
        floor((3*n-1)/2) ;
    elif k <0 or k > n then
        0;
    else
        procname(n-1,k-1)+procname(n-1,k) ;
    end if;
end proc: # R. J. Mathar, Apr 26 2015

T[n_, k_] := T[n, k] = Which[k == 0 || k == n, 1, k == 1 || k == n-1, Floor[(3n-1)/2], k < 0 || k > n, 0, True, T[n-1, k-1] + T[n-1, k]];
Table[T[n, k], {n, 0, 11}, {k, 0, n}] // Flatten (* Jean-François Alcover, Jan 30 2018 *)

def T(n,k): # T = A026637
    if k==0 or k==n: return 1
    elif k==1 or k==n-1: return ((3*n-1)//2)
    else: return T(n-1, k) + T(n-1, k-1)
flatten([[T(n,k) for k in range(n+1)] for n in range(16)]) # G. C. Greubel, Jun 28 2024

From G. C. Greubel, Jun 28 2024: (Start)
T(n, n-k) = T(n, k).
T(2*n-1, n-1) = A026641(n), n >= 1.
Sum_{k=0..n} T(n, k) = A026644(n).
Sum_{k=0..n} (-1)^k*T(n, k) = A000007(n). (End)

A172061 Expansion of (2/(3sqrt(1-4z)-1+4z))((1-sqrt(1-4z))/(2z))^k with k=4.

Original entry on oeis.org

1, 5, 22, 91, 367, 1461, 5776, 22748, 89402, 350974, 1377174, 5403193, 21201211, 83211277, 326703424, 1283211208, 5042294926, 19822108582, 77958648604, 306739666198, 1207433301046, 4754874514690, 18732340230592, 73827134976216

Offset: 0

Richard Choulet, Jan 24 2010

This sequence is the 4th diagonal below the main diagonal (which itself is A026641) in the array which grows with "Pascal rule" given here by rows: 1,0,1,0,1,0,1,0,1,0,1,0,1,0, 1,1,1,1,1,1,1,1,1,1,1,1,1,1, 1,1,2,2,3,3,4,4,5,5,6,6,7,7, 1,2,4,6,9,12,16,20,25,30, 1,3,7,13,22,34,50,70,95. The MAPLE programs give the first diagonals of this array.

Apparently the number of peaks in all Dyck paths of semilength n+4 that are 2 steps higher than the preceding peak. - David Scambler, Apr 22 2013

			a(4) = C(12,4)-C(11,3)+C(10,2)-C(9,1)+C(8,0)=55*9-55*3+45-9+1=367.

Vincenzo Librandi, Table of n, a(n) for n = 0..200

Cf. A091526 (k=-2), A072547 (k=-1), A026641 (k=0), A014300 (k=1), A014301 (k=2), A172025 (k=3), A172062 (k=5), A172063 (k=6), A172064 (k=7), A172065 (k=8), A172066 (k=9), A172067 (k=10).

k:=4; m:=30; R:=PowerSeriesRing(Rationals(), m); Coefficients(R!( (2/(3*Sqrt(1-4*x)-1+4*x))*((1-Sqrt(1-4*x))/(2*x))^k )); // G. C. Greubel, Feb 16 2019

a:= n-> add((-1)^(p)*binomial(2*n+4-p, n-p), p=0..n):
seq(a(n), n=0..30);
# second Maple program:
gf:= (2/(3*sqrt(1-4*z)-1+4*z))*((1-sqrt(1-4*z))/(2*z))^4:
a:= n-> coeff(series(gf, z, n+10), z, n):
seq(a(n), n=0..30);

CoefficientList[Series[(2/(3*Sqrt[1-4*x]-1+4*x))*((1-Sqrt[1-4*x])/(2*x))^4, {x, 0, 20}], x] (* Vaclav Kotesovec, Apr 19 2014 *)

k=4; my(x='x+O('x^30)); Vec((2/(3*sqrt(1-4*x)-1+4*x))*((1-sqrt(1-4*x))/(2*x))^k) \\ G. C. Greubel, Feb 16 2019

k=4; ((2/(3*sqrt(1-4*x)-1+4*x))*((1-sqrt(1-4*x))/(2*x))^k).series(x, 20).coefficients(x, sparse=False) # G. C. Greubel, Feb 16 2019

G.f.: (2/(3*sqrt(1-4*x)-1+4*x))*((1-sqrt(1-4*x))/(2*x))^k with k=4.
a(n) = Sum_{p=0..n} (-1)^(p)*binomial(2*n+k-p, n-p), with k=4.
a(n) ~ 2^(2*n+5)/(3*sqrt(Pi*n)). - Vaclav Kotesovec, Apr 19 2014
D-finite with recurrence: +2*(n+4)*a(n) +(-13*n-36)*a(n-1) +(15*n+16)*a(n-2) +(19*n+14)*a(n-3) +2*(2*n-1)*a(n-4)=0. - R. J. Mathar, Feb 21 2020

A091526 Coefficient of x^n in 1/((1+x)*(1-x)^(n-1)).

Original entry on oeis.org

1, -1, 1, 2, 9, 34, 130, 496, 1897, 7274, 27966, 107788, 416394, 1611908, 6251596, 24287212, 94499689, 368202778, 1436458486, 5610483532, 21936442894, 85852554748, 336300861436, 1318441228432, 5172792817834, 20309402206084

Offset: 0

Michael Somos, Jan 18 2004

sign

Number of positive terms in expansion of (x_1+x_2+...+x_{n-1}-x_n)^(n+1). - Sergio Falcon, Feb 08 2007

Without the beginning "1" and "-1", we obtain the second diagonal over the principal diagonal of the array notified by B. Cloitre in A026641 and used by R. Choulet in A172025, and from A172061 to A172066. - Richard Choulet, Jan 25 2010

Vincenzo Librandi, Table of n, a(n) for n = 0..200

Cf. A172025, A172061-A172066. - Richard Choulet, Jan 25 2010

Cf. A072547 (k=-1), A026641 (k=0), A014300 (k=1), A014301 (k=2), A172025 (k=3), A172061 (k=4), A172062 (k=5), A172063 (k=6), A172064 (k=7), A172065 (k=8), A172066 (k=9), A172067 (k=10).

k:=-2; m:=30; R:=PowerSeriesRing(Rationals(), m); Coefficients(R!( (2/(3*Sqrt(1-4*x)-1+4*x))*((1-Sqrt(1-4*x))/(2*x))^k )); // G. C. Greubel, Feb 18 2019

for n from 0 to 40 do a(n):=sum('(-1)^(p)*binomial(2*n-p+2,2+n-p)',p=0..n+2): od:seq(a(n),n=0..40):od; taylor((2/(3*sqrt(1-4*z)-1+4*z))*((1-sqrt(1-4*z))/(2*z))^(-2),z=0,42); # Richard Choulet, Jan 25 2010

Table[Sum[Binomial[n+i-2, i]*(-1)^(n-i),{i,0,n}],{n,0,20}] (* Vaclav Kotesovec, Apr 19 2014 *)
Table[(-1)^n 2^(1-n)+Binomial[-1+2 n,1+n] Hypergeometric2F1[1,2 n,2+n,-1],{n,0,20}] (* Vaclav Kotesovec, Apr 19 2014 *)
With[{k = -2}, CoefficientList[Series[(2/(3*Sqrt[1-4*x]-1+4*x))*((1 - Sqrt[1-4*x])/(2*x))^k, {x, 0, 30}], x]] (* G. C. Greubel, Feb 18 2019 *)

a(n)=sum(i=0,n,binomial(n+i-2,i)*(-1)^(n-i));

k=-2; my(x='x+O('x^30)); Vec((2/(3*sqrt(1-4*x)-1+4*x))*((1-sqrt(1-4*x))/(2*x))^k) \\ G. C. Greubel, Feb 18 2019

k=-2; ((2/(3*sqrt(1-4*x)-1+4*x))*((1-sqrt(1-4*x))/(2*x))^k).series(x, 30).coefficients(x, sparse=False) # G. C. Greubel, Feb 18 2019

From Richard Choulet, Jan 25 2010: (Start)
G.f: f such as: f(z)=(2/(3*sqrt(1-4*z)-1+4*z))*((1-sqrt(1-4*z))/(2*z))^(-2).
a(n) = Sum_{j=0..n+2} (-1)^j*binomial(2*n-j+2, 2+n-j). (End)
Recurrence: 2*n*(3*n-7)*a(n) = (21*n^2 - 61*n + 48)*a(n-1) + 2*(2*n-3)*(3*n-4)*a(n-2). - Vaclav Kotesovec, Apr 19 2014
a(n) ~ 2^(2*n-1)/(3*sqrt(Pi*n)). - Vaclav Kotesovec, Apr 19 2014

A147855 G.f.: 1 / (1 + 4xG(x)^2 - 7xG(x)^3) where G(x) = 1 + x*G(x)^4 is the g.f. of A002293.

Original entry on oeis.org

1, 3, 22, 174, 1444, 12323, 107104, 942952, 8381596, 75053100, 676017962, 6118171326, 55591175956, 506805088026, 4633571685968, 42468065811884, 390071875757852, 3589637747968964, 33089300640166360, 305476314574338648, 2823932709938708824, 26137341654281261347

Offset: 0

Paul D. Hanna, Jun 16 2013

nonn

			G.f.: A(x) = 1 + 3*x + 22*x^2 + 174*x^3 + 1444*x^4 + 12323*x^5 +...
A related series is G(x) = 1 + x*G(x)^4, where
G(x) = 1 + x + 4*x^2 + 22*x^3 + 140*x^4 + 969*x^5 + 7084*x^6 +...
G(x)^2 = 1 + 2*x + 9*x^2 + 52*x^3 + 340*x^4 + 2394*x^5 + 17710*x^6 +...
G(x)^3 = 1 + 3*x + 15*x^2 + 91*x^3 + 612*x^4 + 4389*x^5 + 32890*x^6 +...
such that A(x) = 1/(1 + 4*x*G(x)^2 - 7*x*G(x)^3).

G. C. Greubel, Table of n, a(n) for n = 0..1000

Cf. A226705, A002293.
Cf. A026641, A183160, A371753.
Cf. A005810, A078995, A226733, A226761, A385605, A386811.

Table[Sum[Binomial[2*n+k,n-k]*Binomial[2*n-k,k],{k,0,n}],{n,0,20}] (* Vaclav Kotesovec, Jun 16 2013 *)

{a(n)=sum(k=0, n, binomial(2*n+k, n-k)*binomial(2*n-k, k))}
for(n=0, 30, print1(a(n), ", "))

{a(n)=sum(k=0, n, binomial(k, n-k)*binomial(4*n-k, k))}
for(n=0, 30, print1(a(n), ", "))

{a(n)=local(G=1+x); for(i=0, n,G=1+x*G^4+x*O(x^n)); polcoeff(1/(1+4*x*G^2-7*x*G^3), n)}
for(n=0, 30, print1(a(n), ", "))

{a(n)=local(G=1+x); for(i=0, n,G=1+x*G^4+x*O(x^n)); polcoeff(1/(1-3*x*G^2-7*x^2*G^6), n)}
for(n=0, 30, print1(a(n), ", "))

a(n) = Sum_{k=0..n} C(k, n-k) * C(4*n-k, k).
a(n) = Sum_{k=0..n} C(n+k, n-k) * C(3*n-k, k).
a(n) = Sum_{k=0..n} C(2*n+k, n-k) * C(2*n-k, k).
a(n) = Sum_{k=0..n} C(3*n+k, n-k) * C(n-k, k).
a(n) = Sum_{k=0..n} C(4*n+k, n-k) * C(-k, k).
G.f.: 1 / (1 - 3*x*G(x)^2 - 7*x^2*G(x)^6) where G(x) = 1 + x*G(x)^4 is the g.f. of A002293.
a(n) ~ 2^(8*n+5/2)/(5*sqrt(Pi*n)*3^(3*n+1/2)). - Vaclav Kotesovec, Jun 16 2013
From Seiichi Manyama, Apr 05 2024: (Start)
a(n) = Sum_{k=0..floor(n/2)} binomial(4*n-2*k-1,n-2*k).
a(n) = [x^n] 1/((1-x^2) * (1-x)^(3*n)). (End)
From Seiichi Manyama, Aug 05 2025: (Start)
a(n) = Sum_{k=0..n} (-2)^(n-k) * binomial(4*n+1,k).
a(n) = Sum_{k=0..n} (-1)^(n-k) * binomial(3*n+k,k). (End)
From Seiichi Manyama, Aug 14 2025: (Start)
a(n) = Sum_{k=0..n} (-1)^k * 2^(n-k) * binomial(4*n+1,k) * binomial(4*n-k,n-k).
G.f.: G(x)^2/((-1+2*G(x)) * (4-3*G(x))) where G(x) = 1+x*G(x)^4 is the g.f. of A002293. (End)
G.f.: B(x)^2/(1 + 5*(B(x)-1)/4), where B(x) is the g.f. of A005810. - Seiichi Manyama, Aug 15 2025

A172025 Expansion of (2/(3sqrt(1-4z)-1+4z))((1-sqrt(1-4z))/(2z))^k with k=3.

Original entry on oeis.org

1, 4, 16, 62, 239, 920, 3544, 13672, 52834, 204528, 793092, 3080226, 11980667, 46662704, 181971248, 710454896, 2776717742, 10863073784, 42537035408, 166704021596, 653827252022, 2566222449104, 10079023179536, 39611016586832

Offset: 0

Richard Choulet, Jan 23 2010

This sequence is the third diagonal below the main diagonal (which itself is A026641) in the array which grows with "Pascal rule" given here by rows:
  1,  0,  1,  0,  1,  0,  1,  0,  1,  0,  1,  0,  1,  0,
  1,  1,  1,  1,  1,  1,  1,  1,  1,  1,  1,  1,  1,  1,
  1,  1,  2,  2,  3,  3,  4,  4,  5,  5,  6,  6,  7,  7,
  1,  2,  4,  6,  9, 12, 16, 20, 25, 30,
  1,  3,  7, 13, 22, 34, 50, 70, 95.
The Maple programs give the first diagonals of this array.
Apparently the number of peaks in all Dyck paths of semilength n+3 that are 1 step higher than the preceding peak. - David Scambler, Apr 22 2013

			a(4) = C(11,4) - C(10,3) + C(9,2) - C(8,1) + C(7,0) = 330 - 120 + 36 - 8 + 1 = 239.

Vincenzo Librandi, Table of n, a(n) for n = 0..1000

Cf. A091526 (k=-2), A072547 (k=-1), A026641 (k=0), A014300 (k=1), A014301 (k=2), A172061 (k=4), A172062 (k=5), A172063 (k=6), A172064 (k=7), A172065 (k=8), A172066 (k=9), A172067 (k=10).

k:=3; m:=30; R:=PowerSeriesRing(Rationals(), m); Coefficients(R!( (2/(3*Sqrt(1-4*x)-1+4*x))*((1-Sqrt(1-4*x))/(2*x))^k )); // G. C. Greubel, Feb 16 2019

a:= n-> add((-1)^(p)*binomial(2*n+3-p,n-p), p=0..n):
seq(a(n), n=0..30);
# second Maple program:
gf:= (2/(3*sqrt(1-4*z)-1+4*z))*((1-sqrt(1-4*z))/(2*z))^3:
a:= n-> coeff(series(gf,z,n+10),z,n):
seq(a(n), n=0..30);

a[n_] := Binomial[2*n+3, n+3]*Hypergeometric2F1[1, -n, -3-2*n, -1]; Table[a[n], {n, 0, 23}] (* Jean-François Alcover, Dec 17 2013 *)

k=3; my(x='x+O('x^30)); Vec((2/(3*sqrt(1-4*x)-1+4*x))*((1-sqrt(1-4*x))/(2*x))^k) \\ G. C. Greubel, Feb 16 2019

k=3; ((2/(3*sqrt(1-4*x)-1+4*x))*((1-sqrt(1-4*x))/(2*x))^k).series(x, 20).coefficients(x, sparse=False) # G. C. Greubel, Feb 16 2019

G.f.: (2/(3*sqrt(1-4*x)-1+4*x))*((1-sqrt(1-4*x))/(2*x))^k with k=3.
a(n) = Sum_{p=0..n} (-1)^(p)*binomial(2*n+k-p,n-p), with k=3.
a(n) ~ 2^(2*n+4)/(3*sqrt(Pi*n)). - Vaclav Kotesovec, Apr 19 2014
Conjecture: 2*n*(n+3)*a(n) + (-7*n^2 - 17*n - 8)*a(n-1) -2*(n+2)*(2*n+1)*a(n-2) = 0. - R. J. Mathar, Feb 19 2016
a(n) = [x^n] 1/((1 - x^2)*(1 - x)^(n+3)). - Ilya Gutkovskiy, Oct 25 2017

A172062 Expansion of (2/(3sqrt(1-4z)-1+4z))((1-sqrt(1-4z))/(2z))^k with k=5.

Original entry on oeis.org

1, 6, 29, 128, 541, 2232, 9076, 36568, 146446, 584082, 2322967, 9220544, 36548573, 144732176, 572756312, 2265577184, 8959034798, 35421613196, 140035644602, 553606049024, 2188652065586, 8653317051056, 34216118389384

Offset: 0

Richard Choulet, Jan 24 2010

This sequence is the 5th diagonal below the main diagonal (which itself is A026641) in the array which grows with "Pascal rule" given here by rows: 1,0,1,0,1,0,1,0,1,0,1,0,1,0, 1,1,1,1,1,1,1,1,1,1,1,1,1,1, 1,1,2,2,3,3,4,4,5,5,6,6,7,7, 1,2,4,6,9,12,16,20,25,30, 1,3,7,13,22,34,50,70,95. The Maple programs give the first diagonals of this array.
Apparently the number of peaks in all Dyck paths of semilength n+5 that are 3 steps higher than the preceding peak. - David Scambler, Apr 22 2013
Apparently half the sum of all height differences between adjacent peaks in all Dyck paths of semilength n+3. - David Scambler, Apr 22 2013

			a(4) = C(13,4) - C(12,3) + C(11,2) - C(10,1) + C(9,0) = 13*11*5 - 20*11 + 55 - 10 + 1 = 541.

Vincenzo Librandi, Table of n, a(n) for n = 0..200

Cf. A091526 (k=-2), A072547 (k=-1), A026641 (k=0), A014300 (k=1), A014301 (k=2), A172025 (k=3), A172061 (k=4), A172063 (k=6), A172064 (k=7), A172065 (k=8), A172066 (k=9), A172067 (k=10).

k:=5; m:=30; R:=PowerSeriesRing(Rationals(), m); Coefficients(R!( (2/(3*Sqrt(1-4*x)-1+4*x))*((1-Sqrt(1-4*x))/(2*x))^k )); // G. C. Greubel, Feb 17 2019

for k from 0 to 20 do for n from 0 to 40 do a(n):=sum('(-1)^(p)*binomial(2*n-p+k, n-p)', p=0..n): od:seq(a(n), n=0..40):od;
# 2nd program
for k from 0 to 40 do taylor((2/(3*sqrt(1-4*z)-1+4*z))*((1-sqrt(1-4*z))/(2*z))^k, z=0, 40+k):od;

CoefficientList[Series[(2/(3*Sqrt[1-4*x]-1+4*x))*((1-Sqrt[1-4*x])/(2*x))^5, {x, 0, 20}], x] (* Vaclav Kotesovec, Apr 19 2014 *)

k=5; my(x='x+O('x^30)); Vec((2/(3*sqrt(1-4*x)-1+4*x))*((1-sqrt(1-4*x))/(2*x))^k) \\ G. C. Greubel, Feb 17 2019

k=5; ((2/(3*sqrt(1-4*x)-1+4*x))*((1-sqrt(1-4*x))/(2*x))^k).series(x, 30).coefficients(x, sparse=False) # G. C. Greubel, Feb 17 2019

a(n) = Sum_{j=0..n} (-1)^j*binomial(2*n+k-j, n-j), with k=5.
a(n) ~ 2^(2*n+6)/(3*sqrt(Pi*n)). - Vaclav Kotesovec, Apr 19 2014
Conjecture: 2*n*(n+5)*(3*n+7)*a(n) - (n+3)*(21*n^2+79*n+80)*a(n-1) - 2*(3*n+10)*(2*n+3)*(n+2)*a(n-2) = 0. - R. J. Mathar, Feb 19 2016

A172063 Expansion of (2/(3sqrt(1-4z)-1+4z))((1-sqrt(1-4z))/(2z))^k with k=6.

Original entry on oeis.org

1, 7, 37, 174, 771, 3300, 13820, 57044, 233108, 945793, 3817351, 15347362, 61520899, 246052888, 982365976, 3916739872, 15599504614, 62076995998, 246866382826, 981218764540, 3898442536366, 15483778158792, 61482966826992

Offset: 0

Richard Choulet, Jan 24 2010

This sequence is the 6th diagonal below the main diagonal (which itself is A026641) in the array which grows with "Pascal rule" given here by rows: 1,0,1,0,1,0,1,0,1,0,1,0,1,0, 1,1,1,1,1,1,1,1,1,1,1,1,1,1, 1,1,2,2,3,3,4,4,5,5,6,6,7,7, 1,2,4,6,9,12,16,20,25,30, 1,3,7,13,22,34,50,70,95. The Maple programs give the first diagonals of this array.

Apparently the number of peaks in all Dyck paths of semilength n+6 that are 4 steps higher than the preceding peak. - David Scambler, Apr 22 2013

			a(4) = C(14,4) - C(13,3) + C(12,2) - C(11,1) + C(10,0) = 7*13*11 - 26*11 + 66 - 11 + 1 = 771.

Vincenzo Librandi, Table of n, a(n) for n = 0..200

Cf. A091526 (k=-2), A072547 (k=-1), A026641 (k=0), A014300 (k=1), A014301 (k=2), A172025 (k=3), A172061 (k=4), A172062 (k=5), A172064 (k=7), A172065 (k=8), A172066 (k=9), A172067 (k=10).

k:=6; m:=30; R:=PowerSeriesRing(Rationals(), m); Coefficients(R!( (2/(3*Sqrt(1-4*x)-1+4*x))*((1-Sqrt(1-4*x))/(2*x))^k )); // G. C. Greubel, Feb 17 2019

for k from 0 to 20 do for n from 0 to 40 do a(n):=sum('(-1)^(p)*binomial(2*n-p+k, n-p)', p=0..n): od:seq(a(n), n=0..40):od;
# 2nd program
for k from 0 to 40 do taylor((2/(3*sqrt(1-4*z)-1+4*z))*((1-sqrt(1-4*z))/(2*z))^k, z=0, 40+k):od;

CoefficientList[Series[(2/(3*Sqrt[1-4*x]-1+4*x))*((1-Sqrt[1-4*x])/(2*x))^6, {x, 0, 20}], x] (* Vaclav Kotesovec, Apr 19 2014 *)

k=6; my(x='x+O('x^30)); Vec((2/(3*sqrt(1-4*x)-1+4*x))*((1-sqrt(1-4*x))/(2*x))^k) \\ G. C. Greubel, Feb 17 2019

k=6; ((2/(3*sqrt(1-4*x)-1+4*x))*((1-sqrt(1-4*x))/(2*x))^k).series(x, 20).coefficients(x, sparse=False) # G. C. Greubel, Feb 17 2019

a(n) = Sum_{j=0..n} (-1)^j * binomial(2*n+k-j, n-j), with k=6.
a(n) ~ 2^(2*n+7)/(3*sqrt(Pi*n)). - Vaclav Kotesovec, Apr 19 2014
Conjecture: 2*n*(n+6)*(n+3)*a(n) -(7*n^3+59*n^2+166*n+160)*a(n-1) -2*(2*n+5)*(n+4)*(n+2)*a(n-2)=0. - R. J. Mathar, Feb 19 2016

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A035317 Pascal-like triangle associated with A000670.

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A014301 Number of internal nodes of even outdegree in all ordered rooted trees with n edges.

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A183160 a(n) = Sum_{k=0..n} C(n+k,n-k)*C(2*n-k,k).

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A026637 Triangular array T read by rows: T(n,0) = T(n,n) = 1 for n >= 0, T(n,1) = T(n,n-1) = floor((3*n-1)/2) for n >= 1, otherwise T(n,k) = T(n-1,k-1) + T(n-1,k) for 2 <= k <= n-2, n >= 4.

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A172061 Expansion of (2/(3*sqrt(1-4*z)-1+4*z))*((1-sqrt(1-4*z))/(2*z))^k with k=4.

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A091526 Coefficient of x^n in 1/((1+x)*(1-x)^(n-1)).

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A183160 a(n) = Sum_{k=0..n} C(n+k,n-k)C(2n-k,k).

A172061 Expansion of (2/(3sqrt(1-4z)-1+4z))((1-sqrt(1-4z))/(2z))^k with k=4.

A147855 G.f.: 1 / (1 + 4xG(x)^2 - 7xG(x)^3) where G(x) = 1 + x*G(x)^4 is the g.f. of A002293.

A172025 Expansion of (2/(3sqrt(1-4z)-1+4z))((1-sqrt(1-4z))/(2z))^k with k=3.

A172062 Expansion of (2/(3sqrt(1-4z)-1+4z))((1-sqrt(1-4z))/(2z))^k with k=5.

A172063 Expansion of (2/(3sqrt(1-4z)-1+4z))((1-sqrt(1-4z))/(2z))^k with k=6.