A026016 -id:A026016 - cp's OEIS frontend

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.

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

A026022 Triangular array T read by rows: T(n,0) = 1 for n >= 0; T(n,k) = C(n,k) for k = 1,2,...,n, for n = 1,2,3; and for n >= 4, T(n,k) = T(n-1,k-1) + T(n-1,k) for k = 1,2,...,[ (n+2)/2 ] and T(n,(n+3)/2) = T(n-1,(n+1)/2) if n is odd.

Original entry on oeis.org

1, 1, 1, 1, 2, 1, 1, 3, 3, 1, 1, 4, 6, 4, 1, 5, 10, 10, 4, 1, 6, 15, 20, 14, 1, 7, 21, 35, 34, 14, 1, 8, 28, 56, 69, 48, 1, 9, 36, 84, 125, 117, 48, 1, 10, 45, 120, 209, 242, 165, 1, 11, 55, 165, 329, 451, 407, 165, 1, 12, 66, 220, 494, 780, 858, 572, 1, 13, 78, 286, 714, 1274, 1638, 1430, 572

Offset: 1

Clark Kimberling

			From _Philippe Deléham_, Mar 12 2013: (Start)
Triangle begins:
1
1, 1
1, 2, 1
1, 3, 3, 1
1, 4, 6, 4
1, 5, 10, 10, 4
1, 6, 15, 20, 14
1, 7, 21, 35, 34, 14
1, 8, 28, 56, 69, 48
1, 9, 36, 84, 125, 117, 48
1, 10, 45, 120, 209, 242, 165
1, 11, 55, 165, 329, 451, 407, 165
Pentagon arithmetic of Delannoy (in E. Lucas):
1, 1, 1, 1, 0
1, 2, 3, 4, 4, 0
1, 3, 6, 10, 14, 14, 0
1, 4, 10, 20, 34, 48, 48, 0
1, 5, 15, 35, 69, 117, 165, 165,
1, 6, 21, 56, 125, 242, 407, 572,
1, 7, 28, 84, 209, 451, 858, 1430  (End)

E. Lucas, Théorie des Nombres, Albert Blanchard, Paris, 1958,tome1, p.88

{T(n, k) = if( 2*k < n+4, binomial( n, k) - binomial( n, k-4), 0)} /* Michael Somos, Jan 08 2012 */

T(n, k) = C(n, k) - C(n, k-4). - Ralf Stephan, Jan 09 2005
T(2n,n) = A026029(n). - Philippe Deléham, Mar 12 2013
T(2n-1,n) = A026016(n), n>0. - Philippe Deléham, Mar 12 2013

A099376 An inverse Chebyshev transform of x^3.

Original entry on oeis.org

0, 1, 4, 14, 48, 165, 572, 2002, 7072, 25194, 90440, 326876, 1188640, 4345965, 15967980, 58929450, 218349120, 811985790, 3029594040, 11338026180, 42550029600, 160094486370, 603784920024, 2282138106804, 8643460269248

Offset: 0

Paul Barry, Oct 13 2004

The sequence is 0,0,0,1,0,4,0,14,0,...with zeros restored. Second binomial transform of (-1)^n*A003518(n). Second binomial transform of expansion of x^3*c(-x)^8, where c(x) is g.f. of A000108. The g.f. is transformed to x^3 under the Chebyshev transformation A(x) -> (1/(1+x^2))*A(x/(1+x^2)). For a sequence b(n), this corresponds to taking Sum_{k=0..floor(n/2)} C(n-k,k) * (-1)^k * b(n-2k), or Sum_{k=0..n} C((n+k)/2,k) * b(k) * (-1)^((n-k)/2) * (1+(-1)^(n-k))/2.

Let X_n be the set of all noncrossing set partitions of an n-element set which either do not contain {n-1,n} as a block, or which do not contain the block {n} whenever 1 and n-1 are in the same block. For n>0, (-1)^n*a(n) gives the value of the Möbius function of X_{n+2} ordered by dual refinement between the discrete and the full partition. For example, X_3 is a chain consisting of 3 elements and its Möbius function between least and greatest element therefore takes the value a(1)=0. - Henri Mühle, Jan 10 2017

G. C. Greubel, Table of n, a(n) for n = 0..1000
H. Mühle, Two Posets of Noncrossing Partitions Coming From Undesired Parking Spaces, arXiv:1701.02109 [math.CO], 2017.

Cf. A000108, A000245, A003518, A071718.

Partial sums of A026016.

[Catalan(n+2) -2*Catalan(n+1): n in [0..30]]; // G. C. Greubel, May 05 2021

Table[CatalanNumber[n+2] -2CatalanNumber[n+1], {n, 0, 30}] (* or *)
Table[4 Binomial[2#+3, #]/(#+4) &[n-1], {n, 0, 30}] (* Michael De Vlieger, Jan 10 2017, latter after Harvey P. Dale at A002057 *)

{a(n)= if(n<1, 0, n++; 2* binomial(2*n, n-2)/n)} /* Michael Somos, Apr 11 2007 */

[catalan_number(n+2) -2*catalan_number(n+1) for n in (0..30)] # G. C. Greubel, May 05 2021

G.f.: (1-2*x)^4*( sqrt((1+2*x)/(1-2*x)) - 1)^8/(256*x^5).
a(n) = Sum_{k=0..n} (k+1)*C(n, (n-k)/2)*(-1)^k*( C(3, k) -3*C(2, k) +3*C(1, k) -C(0, k) )*(1+(-1)^(n-k))/(n+k+2).
a(n) = A002057(n-1). - Michael Somos, Jul 31 2005
Given an ellipse with eccentricity e and major and minor axis a and b respectively, then ((a-b)/ (a+b))^2 = 1*(e/2)^4 +4*(e/2)^6 +14*(e/2)^8 +48*(e/2)^10 + ... - Michael Somos, Apr 11 2007
E.g.f.: exp(2x)*(Bessel_I(1,2x) - Bessel_I(3,2x)). - Paul Barry, Jun 04 2007
D-finite with recurrence (n+3)*(n-1)*a(n) -2*n*(2*n+1)*a(n-1)=0. - R. J. Mathar, Sep 26 2012
a(n) = A000108(n+2) - 2*A000108(n+1) for n>0. - Henri Mühle, Jan 10 2017, corrected Sep 25 2021
G.f.: ( (1-2*x)*c(x) - (1-x) )/x^2, where c(x) is the gf of A000108. - G. C. Greubel, May 05 2021
From Peter Bala, Aug 30 2023: (Start)
a(n) = 2*n/((n+2)*(n+3)) * binomial(2*n+2, n+1).
a(n) = 2*Sum_{k = 0..n-1} 1/(n+1)*binomial(n+1, k)*binomial(n+1, k+2). (End)

A026020 a(n) = binomial(4n, n) - binomial(4n, n - 3).

Original entry on oeis.org

1, 4, 28, 219, 1804, 15314, 132572, 1163565, 10316924, 92195488, 829016968, 7492106505, 67991427828, 619193535380, 5655829748520, 51794730347745, 475390078267356, 4371917301657488, 40276635724273936

Offset: 0

Clark Kimberling

nonn

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

a(n) = T(4n, n), where T is the array defined in A026009.
Bisections are A026012 and A026016.
Cf. A004333, A005810.

[Binomial(4*n, n) - Binomial(4*n, n-3): n in [0..20]]; // G. C. Greubel, Mar 22 2021

A026020:= n-> binomial(4*n,n) - binomial(4*n,n-3); seq(A026020(n), n=0..20); # G. C. Greubel, Mar 22 2021

Table[Binomial[4n, n] - Binomial[4n, n - 3], {n, 0, 19}] (* Alonso del Arte, Jun 06 2019 *)

a(n) = binomial(4*n, n) - binomial(4*n, n-3) \\ Felix Fröhlich, Jun 06 2019

[binomial(4*n, n) - binomial(4*n, n-3) for n in (0..20)] # G. C. Greubel, Mar 22 2021

G.f.: (g - 2)*(1 - g + g^2)*g/(3*g - 4) where g = 1 + x*g^4 is the g.f. of A002293. - Mark van Hoeij, Nov 11 2011

a(n) = A005810(n) - A004333(n) for n > 2 - Felix Fröhlich, Jun 06 2019

A071725 Expansion of (1+x^2C^4)C, where C = (1 - sqrt(1-4x))/(2x) is g.f. for Catalan numbers, A000108.

Original entry on oeis.org

1, 1, 3, 10, 34, 117, 407, 1430, 5070, 18122, 65246, 236436, 861764, 3157325, 11622015, 42961470, 159419670, 593636670, 2217608250, 8308432140, 31212003420, 117544456770, 443690433654, 1678353186780, 6361322162444

Offset: 0

N. J. A. Sloane, Jun 06 2002

nonn

a(n) is the number of Dyck (n+3)-paths for which the first downstep followed by an upstep (or by nothing at all) is in position 6. For example, a(2)=3 counts UUUUDdUDDD, UUUDDdUUDD, UUUDDdUDUD (the downstep in position 6 is in small type). - David Callan, Dec 09 2004

Vincenzo Librandi, Table of n, a(n) for n = 0..200
Hanna Mularczyk, Lattice Paths and Pattern-Avoiding Uniquely Sorted Permutations, arXiv:1908.04025 [math.CO], 2019.

Cf. A000108.

Essentially the same as A026016.

[6*((n^2+1)/((n+2)*(n+3)))*Catalan(n): n in [0..30]]; // G. C. Greubel, Mar 23 2021

A000108:= n-> binomial(2*n, n)/(n+1);
A071725:= n-> 6*((n^2+1)/((n+2)*(n+3)))*A000108(n);
seq(A071725(n), n=0..30); # G. C. Greubel, Mar 23 2021

CoefficientList[Series[(1 +x^2((1-Sqrt[1-4x])/(2x))^4)(1-Sqrt[1-4x])/(2x), {x, 0, 30}], x] (* Vaclav Kotesovec, Mar 21 2014 *)

[6*((n^2+1)/((n+2)*(n+3)))*catalan_number(n) for n in (0..30)] # G. C. Greubel, Mar 23 2021

From Paul Barry, Jun 28 2009: (Start)
E.g.f.: exp(2*x)*dif(Bessel_I(1,2*x) - Bessel_I(2,2*x),x);
a(n) = Sum_{k=0..n} ( (-1)^k*2^(n-k)*binomial(n,k)*binomial(k+1,floor(k/2)) ). (End)
(n+31)*(n+3)*a(n) +(n^2-180*n-219)*a(n-1) -10*(2*n-3)*(n-10)*a(n-2) = 0. - R. J. Mathar, Nov 23 2011
a(n) ~ 3*2^(2*n+1)/(sqrt(Pi)*n^(3/2)). - Vaclav Kotesovec, Mar 21 2014
From G. C. Greubel, Mar 23 2021: (Start)
G.f.: (1-5*x+6*x^2 - (1-3*x+2*x^2)*sqrt(1-4*x))/(2*x^3).
E.g.f.: exp(2*x)*(BesselI(0,2*x) -BesselI(1,2*x) +BesselI(2,2*x) -BesselI(3,2*x)).
a(n) = C(n+2) -3*C(n+1) +2*C(n), where C(n) are the Catalan numbers.
a(n) = 6*((n^2+1)/((n+2)*(n+3)))*C(n). (End)

cp's OEIS Frontend

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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A026022 Triangular array T read by rows: T(n,0) = 1 for n >= 0; T(n,k) = C(n,k) for k = 1,2,...,n, for n = 1,2,3; and for n >= 4, T(n,k) = T(n-1,k-1) + T(n-1,k) for k = 1,2,...,[ (n+2)/2 ] and T(n,(n+3)/2) = T(n-1,(n+1)/2) if n is odd.

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A099376 An inverse Chebyshev transform of x^3.

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A026020 a(n) = binomial(4n, n) - binomial(4n, n - 3).

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A071725 Expansion of (1+x^2*C^4)*C, where C = (1 - sqrt(1-4*x))/(2*x) is g.f. for Catalan numbers, A000108.

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Magma

Maple

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A071725 Expansion of (1+x^2C^4)C, where C = (1 - sqrt(1-4x))/(2x) is g.f. for Catalan numbers, A000108.