A119326 -id:A119326 - cp's OEIS frontend

A119359 Central coefficients of number triangle A119326.

0, 1, 1, 7, 31, 106, 386, 1499, 5755, 21886, 83854, 323302, 1248534, 4828916, 18719364, 72711123, 282867795, 1101981430, 4298723990, 16788997874, 65641296578, 256895812108, 1006307847324, 3945185527582, 15478851119966

Offset: 0

Paul Barry, May 16 2006

a(n) = A119326(2n,n+1). A119358(n)-a(n) = A071688(n).

Table[HypergeometricPFQ[{-1/2 - n/2, 1/2 - n/2, 1 - n/2, -n/2}, {1/2, 1/2, 1}, 1] - KroneckerDelta[n], {n, 0, 20}] (* Vladimir Reshetnikov, Oct 04 2016 *)
Table[(2^n Binomial[1/2, (n+1)/2]  + Binomial[n, n/2] Cos[Pi n/2] + n CatalanNumber[n])/2 - KroneckerDelta[n], {n, 0, 20}] (* Vladimir Reshetnikov, Oct 06 2016 *)

G.f.: (1/sqrt(1-4x)+(1/sqrt(1+4x^2)-1)-c(x)+x*c(-x^2))/2, c(x) the g.f. of A000108;
a(n) = (C(2n,n+1)+C((n-1)/2)*sin(Pi*n/2)-2*0^n-2C(n-1,n/2)*sin(Pi*(n-1)/2))/2.
a(n) = hypergeom([-1/2-n/2, 1/2-n/2, 1-n/2, -n/2], [1/2, 1/2, 1], 1) - 0^n. - Vladimir Reshetnikov, Oct 04 2016

A034839 Triangular array formed by taking every other term of each row of Pascal's triangle.

Original entry on oeis.org

1, 1, 1, 1, 1, 3, 1, 6, 1, 1, 10, 5, 1, 15, 15, 1, 1, 21, 35, 7, 1, 28, 70, 28, 1, 1, 36, 126, 84, 9, 1, 45, 210, 210, 45, 1, 1, 55, 330, 462, 165, 11, 1, 66, 495, 924, 495, 66, 1, 1, 78, 715, 1716, 1287, 286, 13

Offset: 0

N. J. A. Sloane

Number of compositions of n having k parts greater than 1. Example: T(5,2)=5 because we have 3+2, 2+3, 2+2+1, 2+1+2 and 1+2+2. Number of binary words of length n-1 having k runs of consecutive 1's. Example: T(5,2)=5 because we have 1010, 1001, 0101, 1101 and 1011. - Emeric Deutsch, Mar 30 2005
From Gary W. Adamson, Oct 17 2008: (Start)
Received from Herb Conn:
Let T = tan x, then
tan x = T
tan 2x = 2T / (1 - T^2)
tan 3x = (3T - T^3) / (1 - 3T^2)
tan 4x = (4T - 4T^3) / (1 - 6T^2 + T^4)
tan 5x = (5T - 10T^3 + T^5) / (1 - 10T^2 + 5T^4)
tan 6x = (6T - 20T^3 + 6T^5) / (1 - 15T^2 + 15T^4 - T^6)
tan 7x = (7T - 35T^3 + 21T^5 - T^7) / (1 - 21T^2 + 35T^4 - 7T^6)
tan 8x = (8T - 56T^3 + 56T^5 - 8T^7) / (1 - 28T^2 + 70T^4 - 28T^6 + T^8)
tan 9x = (9T - 84T^3 + 126T^5 - 36T^7 + T^9) / (1 - 36 T^2 + 126T^4 - 84T^6 + 9T^8)
... To get the next one in the series, (tan 10x), for the numerator add:
9....84....126....36....1 previous numerator +
1....36....126....84....9 previous denominator =
10..120....252...120...10 = new numerator
For the denominator add:
......9.....84...126...36...1 = previous numerator +
1....36....126....84....9.... = previous denominator =
1....45....210...210...45...1 = new denominator
...where numerators = A034867, denominators = A034839
(End)
Triangle, with zeros omitted, given by (1, 0, 1, 0, 0, 0, 0, 0, 0, 0, ...) DELTA (0, 1, -1, 0, 0, 0, 0, 0, 0, 0, ...) where DELTA is the operator defined in A084938. - Philippe Deléham, Dec 12 2011
The row (1,66,495,924,495,66,1) plays a role in expansions of powers of the Dedekind eta function. See the Chan link, p. 534. - Tom Copeland, Dec 12 2016
Binomial(n,2k) is also the number of permutations avoiding both 123 and 132 with k ascents, i.e., positions with w[i]Lara Pudwell, Dec 19 2018
Coefficients in expansion of ((x-1)^n+(x+1)^n)/2 or ((x-i)^n+(x+i)^n)/2 with alternating sign. - Eugeniy Sokol, Sep 20 2020
Number of permutations of length n avoiding simultaneously the patterns 213 and 312 with the maximum number of non-overlapping descents equal k (equivalently, with the maximum number of non-overlapping ascents equal k). An ascent (resp., descent) in a permutation a(1)a(2)...a(n) is position i such that a(i) < a(i+1) (resp., a(i) > a(i+1)). - Tian Han, Nov 16 2023

			Triangular array begins:
  1
  1
  1  1
  1  3
  1  6  1
  1 10  5
  1 15 15 1
  ...
cosh(4x) = (cosh x)^5 + 10 (cosh x)^3 (sinh x)^2 + 5 (cosh x) (sinh x)^4, so row 4 is (1,10,5). See Mathematica program. - _Clark Kimberling_, Aug 03 2024

G. C. Greubel, Table of n, a(n) for the first 101 rows, flattened
M. Bukata, R. Kulwicki, N. Lewandowski, L. Pudwell, J. Roth, and T. Wheeland, Distributions of Statistics over Pattern-Avoiding Permutations, arXiv preprint arXiv:1812.07112 [math.CO], 2018.
H. Chan, S. Cooper, and P. Toh, The 26th power of Dedekind's eta function Advances in Mathematics, 207 (2006) 532-543.
Tom Copeland, Juggling Zeros in the Matrix: Example II, 2020.
C. Corsani, D. Merlini, and R. Sprugnoli, Left-inversion of combinatorial sums, Discrete Mathematics, 180 (1998) 107-122.
Tian Han and Sergey Kitaev, Joint distributions of statistics over permutations avoiding two patterns of length 3, arXiv:2311.02974 [math.CO], 2023.
S.-M. Ma, On some binomial coefficients related to the evaluation of tan(nx), arXiv preprint arXiv:1205.0735 [math.CO], 2012. - From _N. J. A. Sloane_, Oct 13 2012
K. Oliver and H. Prodinger, The continued fraction expansion of Gauss' hypergeometric function and a new application to the tangent function, Transactions of the Royal Society of South Africa, Vol. 76 (2012), 151-154, [DOI]; [PDF]. - From _N. J. A. Sloane_, Jan 03 2013
Eric Weisstein's World of Mathematics, Tangent [From _Eric W. Weisstein_, Oct 18 2008]

Cf. A007318, A034867, A034870, A080928, A119326, A201701.

Cf. A008619 (row lengths), A086645.

/* As a triangle */ [[Binomial(n,2*k):k in [0..Floor(n/2)]] : n in [0..10]]; // G. C. Greubel, Feb 23 2018

for n from 0 to 13 do seq(binomial(n,2*k),k=0..floor(n/2)) od;# yields sequence in triangular form; # Emeric Deutsch, Mar 30 2005

u[1, x_] := 1; v[1, x_] := 1; z = 12;
u[n_, x_] := u[n - 1, x] + x*v[n - 1, x]
v[n_, x_] := u[n - 1, x] + v[n - 1, x]
cu = Table[CoefficientList[u[n, x], x], {n, 1, z}];
TableForm[cu]  (* A034839 as a triangle *)
cv = Table[CoefficientList[v[n, x], x], {n, 1, z}];
TableForm[cv]  (* A034867 as a triangle *)
(* Clark Kimberling, Feb 18 2012 *)
Table[Binomial[n, k], {n, 0, 13}, {k, 0, Floor[n, 2], 2}] // Flatten (* Michael De Vlieger, Dec 13 2016 *)
(* The triangle gives coefficients for cosh(nx) as a linear combination of products (cosh(x)^h)*(sinh(x)^k) *)
Column[Table[TrigExpand[Cosh[n  x]], {n, 0, 10}]]
(* Clark Kimberling, Aug 03 2024 *)

for(n=0,15, for(k=0,floor(n/2), print1(binomial(n, 2*k), ", "))) \\ G. C. Greubel, Feb 23 2018

E.g.f.: exp(x)*cosh(x*sqrt(y)). - Vladeta Jovovic, Mar 20 2005
From Emeric Deutsch, Mar 30 2005: (Start)
T(n, k) = binomial(n, 2*k), for n >= 0 and k = 0, 1, ..., floor(n/2).
G.f.: (1-z)/((1-z)^2 - t*z^2). (End)
O.g.f. for column no. k is (1/(1-x))*(x/(1-x))^(2*k), k >= 0 [from the g.f. given in the preceding formula]. - Wolfdieter Lang, Jan 18 2013
From Peter Bala, Jul 14 2015: (Start)
Stretched Riordan array ( 1/(1 - x ), x^2/(1 - x)^2 ) in the terminology of Corsani et al.
Denote this array by P. Then P * A007318 = A201701.
P * transpose(P) is A119326 read as a square array.
Let Q denote the array ( (-1)^k*binomial(2*n,k) )n,k>=0. Q is a signed version of A034870. Then Q*P = the identity matrix, that is, Q is a left-inverse array of P (see Corsani et al., p. 111).
P * A034870 = A080928. (End)
Even rows are A086645. An aerated version of this array is A099174 with each diagonal divided by the first element of the diagonal, the double factorials A001147. - Tom Copeland, Dec 12 2015

A119358 Number of n-element subsets of [2n] having an even sum.

Original entry on oeis.org

1, 1, 2, 10, 38, 126, 452, 1716, 6470, 24310, 92252, 352716, 1352540, 5200300, 20056584, 77558760, 300546630, 1166803110, 4537543340, 17672631900, 68923356788, 269128937220, 1052049129144, 4116715363800, 16123803193628, 63205303218876, 247959261273752

Offset: 0

Paul Barry, May 16 2006

Old name was: Central coefficients of number triangle A119326.

			a(3) = 10: {1,2,3}, {1,2,5}, {1,3,4}, {1,3,6}, {1,4,5}, {1,5,6}, {2,3,5}, {2,4,6}, {3,4,5}, {3,5,6}. - _Alois P. Heinz_, Feb 04 2017

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

Cf. A071688, A119326, A119359, A169888, A282011.

Column k=2 of A318557.

a:= proc(n) option remember; `if`(n<3, 1+n*(n-1)/2,
     ((4*n-10)*(5*n^2-10*n+4)*(a(n-1)+4*(n-2)*a(n-3)
      /(n-1))/(5*n^2-20*n+19)-4*(n-1)*a(n-2))/n)
    end:
seq(a(n), n=0..30);  # Alois P. Heinz, Aug 26 2018

Table[HypergeometricPFQ[{1/2 - n/2, 1/2 - n/2, -n/2, -n/2}, {1/2, 1/2, 1}, 1], {n, 0, 20}] (* Vladimir Reshetnikov, Oct 04 2016 *)

G.f.: (1/sqrt(1-4x)+1/sqrt(1+4x^2))/2.
a(n) = Sum_{k=0..floor(n/2)} C(n,2k)^2.
a(n) = C(2n,n)/2+sin(Pi*(n+1)/2)*C(n,n/2)/2.
a(n) = A119326(2n,n).
a(n) = A071688(n) + A119359(n) for n>=1.
D-finite with recurrence n*(n-1)*(10*n-29)*a(n) +2*(n-1)*(5*n^2-74*n+164)*a(n-1) +4*(-40*n^3+310*n^2 -744*n+559)*a(n-2) +8*(n-2)*(5*n^2-74*n+164)*a(n-3) -16*(25*n-42)*(n-3)*(2*n-7)*a(n-4)=0. - R. J. Mathar, Nov 05 2012
a(n) = hypergeom([(1-n)/2, (1-n)/2, -n/2, -n/2], [1/2, 1/2, 1], 1). - Vladimir Reshetnikov, Oct 04 2016
a(n) = A282011(2n,n). - Alois P. Heinz, Feb 04 2017

New name from Alois P. Heinz, Feb 04 2017

A119335 Number triangle T(n,k) = Sum_{j=0..n-k} C(k,3j)*C(n-k,3j).

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 5, 5, 1, 1, 1, 1, 1, 1, 11, 17, 11, 1, 1, 1, 1, 1, 1, 21, 41, 41, 21, 1, 1, 1, 1, 1, 1, 36, 81, 101, 81, 36, 1, 1, 1, 1, 1, 1, 57, 141, 201, 201, 141, 57, 1, 1, 1

Offset: 0

Paul Barry, May 14 2006

Row sums are A119336. Product of Pascal's triangle and A119337.

			Triangle begins
1;
1, 1;
1, 1, 1;
1, 1, 1,  1;
1, 1, 1,  1,  1;
1, 1, 1,  1,  1,   1;
1, 1, 1,  2,  1,   1,  1;
1, 1, 1,  5,  5,   1,  1,  1;
1, 1, 1, 11, 17,  11,  1,  1, 1;
1, 1, 1, 21, 41,  41, 21,  1, 1, 1;
1, 1, 1, 36, 81, 101, 81, 36, 1, 1, 1;

Seiichi Manyama, Rows n = 0..139, flattened

T(2n,n) gives A119363.

Cf. A119326.

T[n_, k_] := Sum[Binomial[k, 3j] Binomial[n-k, 3j], {j, 0, n-k}];
Table[T[n, k], {n, 0, 11}, {k, 0, n}] // Flatten (* Jean-François Alcover, Sep 14 2023 *)

Column k has g.f. (x^k/(1-x)) * Sum_{j=0..k} C(k,3j)(x/(1-x))^(3j).

More terms from Seiichi Manyama, Mar 12 2019

A119328 Number triangle T(n,k)=sum{i=0..n, (-1)^(n-i)C(n,i)sum{j=0..i-k, C(k,2j)*C(i-k,2j)}}.

Original entry on oeis.org

1, 0, 1, 0, -1, 1, 0, 1, -2, 1, 0, -1, 4, -3, 1, 0, 1, -6, 9, -4, 1, 0, -1, 8, -19, 16, -5, 1, 0, 1, -10, 33, -44, 25, -6, 1, 0, -1, 12, -51, 96, -85, 36, -7, 1, 0, 1, -14, 73, -180, 225, -146, 49, -8, 1, 0, -1, 16, -99, 304, -501, 456, -231, 64, -9, 1

Offset: 0

Paul Barry, May 14 2006

Row sums are A021913(n+2). Product with Pascal's triangle A007318 is A119326.

			Triangle begins
1,
0, 1,
0, -1, 1,
0, 1, -2, 1,
0, -1, 4, -3, 1,
0, 1, -6, 9, -4, 1,
0, -1, 8, -19, 16, -5, 1,
0, 1, -10, 33, -44, 25, -6, 1,
0, -1, 12, -51, 96, -85, 36, -7, 1,
0, 1, -14, 73, -180, 225, -146, 49, -8, 1,
0, -1, 16, -99, 304, -501, 456, -231, 64, -9, 1

t[n_, k_] := Sum[(-1)^(n - i)*Binomial[n, i]*Sum[Binomial[k, 2 j]*Binomial[i - k, 2 j], {j, 0, i - k}], {i, 0, n}]; Table[t[n, k], {n, 0, 10}, {k, 0, n}] // Flatten (* Jean-François Alcover, Mar 25 2013 *)

Column k has g.f. (x/(1+x))^k*sum{j=0..k, C(k,2j)x^(2j)}

A307090 Number triangle T(n,k) = Sum_{j=0..n-k} (-1)^j * binomial(k,2j) binomial(n-k,2*j).

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 0, 1, 1, 1, 1, -2, -2, 1, 1, 1, 1, -5, -8, -5, 1, 1, 1, 1, -9, -17, -17, -9, 1, 1, 1, 1, -14, -29, -34, -29, -14, 1, 1, 1, 1, -20, -44, -54, -54, -44, -20, 1, 1, 1, 1, -27, -62, -74, -74, -74, -62, -27, 1, 1, 1, 1, -35, -83, -90, -74, -74, -90, -83, -35, 1, 1

Offset: 0

Seiichi Manyama, Mar 24 2019

			Triangle begins:
n\k | 0  1    2    3    4    5    6  7  8
----+-------------------------------------
0   | 1;
1   | 1, 1;
2   | 1, 1,   1;
3   | 1, 1,   1,   1;
4   | 1, 1,   0,   1,   1;
5   | 1, 1,  -2,  -2,   1,   1;
6   | 1, 1,  -5,  -8,  -5,   1,   1;
7   | 1, 1,  -9, -17, -17,  -9,   1, 1;
8   | 1, 1, -14, -29, -34, -29, -14, 1, 1;

Seiichi Manyama, Rows n = 0..139, flattened

Row sums give A099587(n+1).
T(2*n,n) gives A307091.
Cf. A007318, A098593, A119326, A119335.

T[n_, k_] := Sum[(-1)^j * Binomial[k, 2*j] * Binomial[n - k, 2*j], {j, 0, n - k}]; Table[T[n, k], {n, 0, 11}, {k, 0, n}] // Flatten (* Amiram Eldar, May 20 2021 *)

A307156 Number triangle T(n,k) = Sum_{j=0..n-k} (-1)^j * binomial(k,3j) binomial(n-k,3*j).

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 0, 1, 1, 1, 1, 1, 1, -3, -3, 1, 1, 1, 1, 1, 1, -9, -15, -9, 1, 1, 1, 1, 1, 1, -19, -39, -39, -19, 1, 1, 1, 1, 1, 1, -34, -79, -99, -79, -34, 1, 1, 1, 1, 1, 1, -55, -139, -199, -199, -139, -55, 1, 1, 1

Offset: 0

Seiichi Manyama, Mar 27 2019

			Triangle begins:
n\k | 0  1  2   3    4   5  6  7  8
----+-------------------------------
0   | 1;
1   | 1, 1;
2   | 1, 1, 1;
3   | 1, 1, 1,  1;
4   | 1, 1, 1,  1,   1;
5   | 1, 1, 1,  1,   1,  1;
6   | 1, 1, 1,  0,   1,  1, 1;
7   | 1, 1, 1, -3,  -3,  1, 1, 1;
8   | 1, 1, 1, -9, -15, -9, 1, 1, 1;

Seiichi Manyama, Rows n = 0..139, flattened

Row sums give A307089.
T(2*n,n) gives A307158.
Cf. A119326, A119335, A307090.

T[n_, k_] := Sum[(-1)^j * Binomial[k, 3*j] * Binomial[n - k, 3*j], {j, 0, n - k}]; Table[T[n, k], {n, 0, 11}, {k, 0, n}] // Flatten (* Amiram Eldar, May 20 2021 *)

A119327 Expansion of (1-4x+12x^2-16x^3+8x^4)/(1-x)^5.

Original entry on oeis.org

1, 1, 7, 19, 38, 66, 106, 162, 239, 343, 481, 661, 892, 1184, 1548, 1996, 2541, 3197, 3979, 4903, 5986, 7246, 8702, 10374, 12283, 14451, 16901, 19657, 22744, 26188, 30016, 34256, 38937, 44089, 49743, 55931, 62686, 70042, 78034, 86698, 96071

Offset: 0

Paul Barry, May 14 2006

Seiichi Manyama, Table of n, a(n) for n = 0..10000
Index entries for linear recurrences with constant coefficients, signature (5,-10,10,-5,1).

Fifth column of A119326.

LinearRecurrence[{5,-10,10,-5,1},{1,1,7,19,38},41] (* James C. McMahon, Sep 15 2024 *)

N=66; x='x+O('x^N); Vec((1-4*x+12*x^2-16*x^3+8*x^4)/(1-x)^5) \\ Seiichi Manyama, Mar 11 2019

a(n) = Sum{j=0..n} C(4,2j)*C(n,2j).

A119329 Number triangle T(n,k)=sum{j=0..n-k, C(k,2j)C(n-k,2j)*2^j}.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 3, 1, 1, 1, 1, 7, 7, 1, 1, 1, 1, 13, 19, 13, 1, 1, 1, 1, 21, 37, 37, 21, 1, 1, 1, 1, 31, 61, 77, 61, 31, 1, 1, 1, 1, 43, 91, 141, 141, 91, 43, 1, 1, 1, 1, 57, 127, 241, 301, 241, 127, 57, 1, 1

Offset: 0

Paul Barry, May 14 2006

Row sums are A119330. Product of Pascal's triangle A007318 and A119331. T(n,k)=T(n,n-k).

			Triangle begins
1,
1, 1,
1, 1, 1,
1, 1, 1, 1,
1, 1, 3, 1, 1,
1, 1, 7, 7, 1, 1,
1, 1, 13, 19, 13, 1, 1,
1, 1, 21, 37, 37, 21, 1, 1

Cf. A119326.

Column k has g.f. (x^k/(1-x))*sum{j=0..k, C(k,2j)*2^j*(x/(1-x))^(2j)}

cp's OEIS Frontend

A119359 Central coefficients of number triangle A119326.

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A034839 Triangular array formed by taking every other term of each row of Pascal's triangle.

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A119358 Number of n-element subsets of [2n] having an even sum.

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A119335 Number triangle T(n,k) = Sum_{j=0..n-k} C(k,3j)*C(n-k,3j).

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A119328 Number triangle T(n,k)=sum{i=0..n, (-1)^(n-i)*C(n,i)*sum{j=0..i-k, C(k,2j)*C(i-k,2j)}}.

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A307090 Number triangle T(n,k) = Sum_{j=0..n-k} (-1)^j * binomial(k,2*j) * binomial(n-k,2*j).

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A307156 Number triangle T(n,k) = Sum_{j=0..n-k} (-1)^j * binomial(k,3*j) * binomial(n-k,3*j).

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A119327 Expansion of (1-4x+12x^2-16x^3+8x^4)/(1-x)^5.

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A119328 Number triangle T(n,k)=sum{i=0..n, (-1)^(n-i)C(n,i)sum{j=0..i-k, C(k,2j)*C(i-k,2j)}}.

A307090 Number triangle T(n,k) = Sum_{j=0..n-k} (-1)^j * binomial(k,2j) binomial(n-k,2*j).

A307156 Number triangle T(n,k) = Sum_{j=0..n-k} (-1)^j * binomial(k,3j) binomial(n-k,3*j).