A004311 -id:A004311 - cp's OEIS frontend

A092392 Triangle read by rows: T(n,k) = C(2*n - k,n), 0 <= k <= n.

1, 2, 1, 6, 3, 1, 20, 10, 4, 1, 70, 35, 15, 5, 1, 252, 126, 56, 21, 6, 1, 924, 462, 210, 84, 28, 7, 1, 3432, 1716, 792, 330, 120, 36, 8, 1, 12870, 6435, 3003, 1287, 495, 165, 45, 9, 1, 48620, 24310, 11440, 5005, 2002, 715, 220, 55, 10, 1, 184756, 92378, 43758, 19448, 8008, 3003, 1001, 286, 66, 11, 1

Offset: 0

Ralf Stephan, Mar 21 2004

First column is C(2*n,n) or A000984. Central coefficients are C(3*n,n) or A005809. - Paul Barry, Oct 14 2009
T(n,k) = A046899(n,n-k), k = 0..n-1. - Reinhard Zumkeller, Jul 27 2012
From  Peter Bala, Nov 03 2015: (Start)
Viewed as the square array [binomial (2*n + k, n + k)]n,k>=0  this is the generalized Riordan array ( 1/sqrt(1 - 4*x),c(x) ) in the sense of the Bala link, where c(x) is the o.g.f. for A000108.
The square array factorizes as ( 1/(2 - c(x)),x*c(x) ) * ( 1/(1 - x),1/(1 - x) ), which equals the matrix product of A100100 with the square Pascal matrix [binomial (n + k,k)]n,k>=0. See the example below. (End)

			From _Paul Barry_, Oct 14 2009: (Start)
Triangle begins
  1,
  2, 1,
  6, 3, 1,
  20, 10, 4, 1,
  70, 35, 15, 5, 1,
  252, 126, 56, 21, 6, 1,
  924, 462, 210, 84, 28, 7, 1,
  3432, 1716, 792, 330, 120, 36, 8, 1
Production array is
  2, 1,
  2, 1, 1,
  2, 1, 1, 1,
  2, 1, 1, 1, 1,
  2, 1, 1, 1, 1, 1,
  2, 1, 1, 1, 1, 1, 1,
  2, 1, 1, 1, 1, 1, 1, 1,
  2, 1, 1, 1, 1, 1, 1, 1, 1,
  2, 1, 1, 1, 1, 1, 1, 1, 1, 1 (End)
As a square array = A100100 * square Pascal matrix:
  /1   1  1  1 ...\   / 1          \/1 1  1  1 ...\
  |2   3  4  5 ...|   | 1 1        ||1 2  3  4 ...|
  |6  10 15 21 ...| = | 3 2 1      ||1 3  6 10 ...|
  |20 35 56 84 ...|   |10 6 3 1    ||1 4 10 20 ...|
  |70 ...         |   |35 ...      ||1 ...        |
- _Peter Bala_, Nov 03 2015

Reinhard Zumkeller, Rows n=0..149 of triangle, flattened
P. Bala, Notes on generalized Riordan arrays
P. Barry, On the Central Coefficients of Riordan Matrices, Journal of Integer Sequences, 16 (2013), #13.5.1.
Paul Barry, Jacobsthal Decompositions of Pascal's Triangle, Ternary Trees, and Alternating Sign Matrices, Journal of Integer Sequences, 19, 2016, #16.3.5.
Ik-Pyo Kim, Michael J. Tsatsomeros, Inverse Relations in Shapiro's Open Questions, arXiv:1707.06590 [math.CO], 2017. See p. 7.

Rows 0-14 are A000984, A001700, A001791, A002054, A002694, A003516, A002696, A030053, A004310, A030054, A004311, A030055, A004312, A030056, A004313.
Columns are A000217, A000292, A000332, A000389, A000579.
Diagonals are A005809, A045721, A025174, A004319, A013698, A003408.
Cf. A100100.

a092392 n k = a092392_tabl !! (n-1) !! (k-1)
a092392_row n = a092392_tabl !! (n-1)
a092392_tabl = map reverse a046899_tabl
-- Reinhard Zumkeller, Jul 27 2012

/* As a triangle */ [[Binomial(2*n-k, n): k in [0..n]]: n in [0..10]]; // G. C. Greubel, Nov 22 2017

A092392 := proc(n,k)
    binomial(2*n-k,n-k) ;
end proc:
seq(seq(A092392(n,k),k=0..n),n=0..10) ; # R. J. Mathar, Feb 06 2015

Table[Binomial[2 n - k, n], {n, 0, 10}, {k, 0, n}] // Flatten (* Michael De Vlieger, Mar 19 2016 *)

C(x):=(1-sqrt(1-4*x))/2;
A(x,y):=(1/sqrt(1-4*x))/(1-y*C(x));
taylor(A(x,y),y,0,10,x,0,10); /* Vladimir Kruchinin, Mar 19 2016 */

for(n=0,10, for(k=0,n, print1(binomial(2*n - k,n), ", "))) \\ G. C. Greubel, Nov 22 2017

As a number triangle, this is T(n, k) = if(k <= n, C(2*n - k, n), 0). Its row sums are C(2*n + 1, n + 1) = A001700. Its diagonal sums are A176287. - Paul Barry, Apr 23 2005
G.f. of column k: 2^k/[sqrt(1 - 4*x)*(1 + sqrt(1 - 4*x))^k].
As a number triangle, this is the Riordan array (1/sqrt(1 - 4*x), x*c(x)), c(x) the g.f. of A000108. - Paul Barry, Jun 24 2005
G.f.: A(x,y)=1/sqrt(1 - 4*x)/(1-y*x*C(x)), where C(x) is g.f. of Catalan numbers. - Vladimir Kruchinin, Mar 19 2016

Diagonal sums comment corrected by Paul Barry, Apr 14 2010

Offset corrected by R. J. Mathar, Feb 08 2013

A100257 Triangle of expansions of 2^(k-1)*x^k in terms of T(n,x), in descending degrees n of T, with T the Chebyshev polynomials.

Original entry on oeis.org

1, 1, 0, 1, 0, 1, 1, 0, 3, 0, 1, 0, 4, 0, 3, 1, 0, 5, 0, 10, 0, 1, 0, 6, 0, 15, 0, 10, 1, 0, 7, 0, 21, 0, 35, 0, 1, 0, 8, 0, 28, 0, 56, 0, 35, 1, 0, 9, 0, 36, 0, 84, 0, 126, 0, 1, 0, 10, 0, 45, 0, 120, 0, 210, 0, 126, 1, 0, 11, 0, 55, 0, 165, 0, 330, 0, 462, 0, 1, 0, 12, 0, 66, 0, 220, 0

Offset: 0

Ralf Stephan, Nov 13 2004

			x^0 = T(0,x)
x^1 = T(1,x) + 0T(0,x)
2x^2 = T(2,x) + 0T(1,x) + 1T(0,x)
4x^3 = T(3,x) + 0T(2,x) + 3T(1,x) + 0T(0,x)
8x^4 = T(4,x) + 0T(3,x) + 4T(2,x) + 0T(1,x) + 3T(0,x)
16x^5 = T(5,x) + 0T(4,x) + 5T(3,x) + 0T(2,x) + 10T(1,x) + 0T(0,x)

M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards Applied Math. Series 55, 1964 (and various reprintings), p. 795.

Vincenzo Librandi, Table of n, a(n) for n = 0..6104
M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards, Applied Math. Series 55, Tenth Printing, 1972 [alternative scanned copy].
H. J. Brothers, Pascal's Prism: Supplementary Material.
Daniel J. Greenhoe, Frames and Bases: Structure and Design, Version 0.20, Signal Processing ABCs series (2019) Vol. 4, see page 175.
Daniel J. Greenhoe, A Book Concerning Transforms, Version 0.10, Signal Processing ABCs series (2019) Vol. 5, see page 97.
Index entries for sequences related to Chebyshev polynomials.

Without zeros: A008311. Row sums are A011782. Cf. A092392.

Diagonals are (with interleaved zeros) twice A001700, A001791, A002054, A002694, A003516, A002696, A030053, A004310, A030054, A004311, A030055, A004312, A030056, A004313.

a[k_, n_] := If[k == 1, 1, If[EvenQ[n] || k < 0 || n > k, 0, If[n >= k - 1, Binomial[2*Floor[k/2], Floor[k/2]]/2, Binomial[k - 1, Floor[n/2]]]]];
Table[a[k, n], {k, 1, 13}, {n, 1, k}] // Flatten (* Jean-François Alcover, May 04 2017, translated from PARI *)

a(k,n)=if(k==1,1,if(n%2==0||k<0||n>k,0,if(n>=k-1,binomial(2*floor(k/2),floor(k/2))/2,binomial(k-1,floor(n/2)))))

A213347 5-quantum transitions in systems of N>=5 spin 1/2 particles, in columns by combination indices.

Original entry on oeis.org

1, 12, 84, 7, 448, 112, 2016, 1008, 36, 8064, 6720, 720, 29568, 36960, 7920, 165, 101376, 177408, 63360, 3960, 329472, 768768, 411840, 51480, 715, 1025024, 3075072, 2306304, 480480, 20020, 3075072, 11531520, 11531520

Offset: 5

Stanislav Sykora, Jun 13 2012

For a general discussion, please see A213343.
This a(n) is for quintuple-quantum transitions (q = 5).
It lists the flattened triangle T(5;N,k) with rows N = 5,6,... and columns N, k = 0..floor((N-5)/2).

			Starting rows of the triangle:
  N | k = 0, 1, ..., floor((N-5)/2)
  5 |    1
  6 |   12
  7 |   84    7
  8 |  448  112
  9 | 2016 1008 36

See A213343

Stanislav Sykora, Table of n, a(n) for n = 5..2356
Stanislav Sykora, T(5;N,k) with rows N=5,..,100 and columns k=0,..,floor((N-5)/2)
Stanislav Sýkora, Magnetic Resonance on OEIS, Stan's NMR Blog (Dec 31, 2014), Retrieved Nov 12, 2019.

Cf. A051288 (q=0), A213343 to A213346 (q=1 to 4), A213348 to A213352 (q=6 to 10).

A054849 (first column), A004311 (row sums).

With[{q = 5}, Table[2^(n - q - 2 k)*Binomial[n, k] Binomial[n - k, q + k], {n, 15}, {k, 0, Floor[(n - q)/2]}]] // Flatten (* Michael De Vlieger, Nov 18 2019 *)

PARI
```
See A213343; set thisq = 5
```

Set q = 5 in: T(q;N,k) = 2^(N-q-2*k)*binomial(N,k)*binomial(N-k,q+k).

A114253 a(n) = C(5+2n,5+n)C(10+2*n,0+n).

Original entry on oeis.org

1, 84, 3276, 92400, 2187900, 46558512, 923410488, 17439488352, 317907339750, 5644249611000, 98209943231400, 1682207622669600, 28457345616827400, 476607460678020000, 7917519856977720000, 130649634333275016960, 2143941655711783421340, 35018537985874435552560

Offset: 0

Zerinvary Lajos, Feb 04 2006

			If n=1 then C(5+2*1,5+1)*C(10+2*1,0+1) = C(7,6)*C(12,1) = 7*12 = 84.
If n=11 then C(5+2*n,5+n)*C(10+2*n,0+n) = C(27,16)*C(32,11) = 13037895*129024480 = 1682207622669600.

Robert Israel, Table of n, a(n) for n = 0..828

Cf. A003516, A004311, A062190.

seq(binomial(5+2*n,5+n)*binomial(10+2*n,n),n=0..30); # Robert Israel, Jan 11 2019

a[n_] := Binomial[2*n + 5, n + 5]*Binomial[2*n + 10, n]; Array[a, 20, 0] (* Amiram Eldar, Sep 06 2025 *)

From Robert Israel, Jan 11 2019: (Start)
(n+1)^2*(11+n)*a(n+1) = 4*(7+2*n)*(3+n)*(11+2*n)*a(n).
a(n) ~ 32768*16^n/(Pi*n). (End)
a(n) = A003516(n+2) * A004311(n+5). - Amiram Eldar, Sep 06 2025

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A092392 Triangle read by rows: T(n,k) = C(2*n - k,n), 0 <= k <= n.

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A100257 Triangle of expansions of 2^(k-1)*x^k in terms of T(n,x), in descending degrees n of T, with T the Chebyshev polynomials.

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A213347 5-quantum transitions in systems of N>=5 spin 1/2 particles, in columns by combination indices.

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A114253 a(n) = C(5+2*n,5+n)*C(10+2*n,0+n).

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A114253 a(n) = C(5+2n,5+n)C(10+2*n,0+n).