A034870 Even-numbered rows of Pascal's triangle.

1, 1, 2, 1, 1, 4, 6, 4, 1, 1, 6, 15, 20, 15, 6, 1, 1, 8, 28, 56, 70, 56, 28, 8, 1, 1, 10, 45, 120, 210, 252, 210, 120, 45, 10, 1, 1, 12, 66, 220, 495, 792, 924, 792, 495, 220, 66, 12, 1, 1, 14, 91, 364, 1001, 2002, 3003, 3432, 3003, 2002, 1001, 364, 91, 14, 1

Offset: 0

Author

N. J. A. Sloane

Keywords

• nonn
• tabf
• easy

Comments

The sequence of row lengths of this array is [1,3,5,7,9,11,13,...]= A005408(n), n>=0.

Equals X^n * [1,0,0,0,...] where X = an infinite tridiagonal matrix with (1,1,1,...) in the main and subsubdiagonal and (2,2,2,...) in the main diagonal. X also = a triangular matrix with (1,2,1,0,0,0,...) in each column. - Gary W. Adamson, May 26 2008

a(n,m) has the following interesting combinatoric interpretation. Let s(n,m) equal the set of all base-4, n-digit numbers with n-m more 1-digits than 2-digits. For example s(2,1) = {10,01,13,31} (note that numbers like 1 are left-padded with 0's to ensure that they have 2 digits). Notice that #s(2,1) = a(2,1) with # indicating cardinality. This is true in general. a(n,m)=#s(n,m). In words, a(n,m) gives the number of n-digit, base-4 numbers with n-m more 1 digits than 2 digits. A proof is provided in the Links section. - Russell Jay Hendel, Jun 23 2015

Examples

			Triangle begins:
  1;
  1,  2,  1;
  1,  4,  6,   4,   1;
  1,  6, 15,  20,  15,   6,   1;
  1,  8, 28,  56,  70,  56,  28,   8,   1;
  1, 10, 45, 120, 210, 252, 210, 120,  45,  10,  1;
  1, 12, 66, 220, 495, 792, 924, 792, 495, 220, 66, 12, 1;
		

Links

• Reinhard Zumkeller, Rows n=0..150 of triangle, flattened
• Peter Bala, Notes on generalized Riordan arrays
• E. H. M. Brietzke, An identity of Andrews and a new method for the Riordan array proof of combinatorial identities, Discrete Math., 308 (2008), 4246-4262.
• Russell Jay Hendel, Proof that a(n,m) gives the number of n-digit, base-4 numbers with n-m more 1-digits than 2-digits.
• Wolfdieter Lang, First 9 rows.
• Franck Ramaharo, Statistics on some classes of knot shadows, arXiv:1802.07701 [math.CO], 2018.
• Franck Ramaharo, A bracket polynomial for 2-tangle shadows, arXiv:2002.06672 [math.CO], 2020.
• Index entries for triangles and arrays related to Pascal's triangle

Crossrefs

Cf. A007318, A034871.

Cf. A000302 (row sums, powers of 4), alternating row sums are 0, except for n=0 which gives 1.

Cf. A003095, A034839, A080928, A086645.

Programs

Haskell

a034870 n k = a034870_tabf !! n !! k
a034870_row n = a034870_tabf !! n
a034870_tabf = map a007318_row [0, 2 ..]
-- Reinhard Zumkeller, Apr 19 2012, Apr 02 2011

Magma

/* As triangle: */ [[Binomial(n,k): k in [0..n]]: n in [0.. 15 by 2]]; // Vincenzo Librandi, Jul 16 2015

Maple

T := (n,k) -> simplify(GegenbauerC(`if`(kPeter Luschny, May 08 2016

Mathematica

Flatten[Table[Binomial[n,k],{n,0,20,2},{k,0,n}]] (* Harvey P. Dale, Dec 15 2014 *)

Maxima

taylor(1/(1-x*(y+1)^2),x,0,10,y,0,10); /* Vladimir Kruchinin, Nov 22 2020 */

Sage

flatten([[binomial(2*n, k) for k in (0..2*n)] for n in (0..12)]) # G. C. Greubel, Mar 18 2022

Formula

T(n, m) = binomial(2*n, m), 0<= m <= 2*n, 0<=n, else 0.

G.f. for column m=2*k sequence: (x^k)*Pe(k, x)/(1-x)^(2*k+1), k>=0; for column m=2*k-1 sequence (x^k)*Po(k, x)/(1-x)^(2*k), k>=1, with the row polynomials Pe(k, x) := sum(A091042(k, m)*x^m, m=0..k) and Po(k, x) := 2*sum(A091044(k, m)*x^m, m=0..k-1); see also triangle A091043.

From Paul D. Hanna, Apr 18 2012: (Start)

Let A(x) be the g.f. of the flattened sequence, then:

G.f.: A(x) = Sum_{n>=0} x^(n^2) * (1+x)^(2*n).

G.f.: A(x) = Sum_{n>=0} x^n*(1+x)^(2*n) * Product_{k=1..n} (1 - (1+x)^2*x^(4*k-3)) / (1 - (1+x)^2*x^(4*k-1)).

G.f.: A(x) = 1/(1 - x*(1+x)^2/(1 + x*(1-x^2)*(1+x)^2/(1 - x^5*(1+x)^2/(1 + x^3*(1-x^4)*(1+x)^2/(1 - x^9*(1+x)^2/(1 + x^5*(1-x^6)*(1+x)^2/(1 - x^13*(1+x)^2/(1 + x^7*(1-x^8)*(1+x)^2/(1 - ...))))))))), a continued fraction.

(End)

From Peter Bala, Jul 14 2015: (Start)

Denote this array by P. Then P * transpose(P) is the square array ( binomial(2*n + 2*k, 2*k) )n,k>=0, which, read by antidiagonals, is A086645.

Transpose(P) is a generalized Riordan array (1, (1 + x)^2) as defined in the Bala link.

Let p(x) = (1 + x)^2. P^2 gives the coefficients in the expansion of the polynomials ( p(p(x)) )^n, P^3 gives the coefficients in the expansion of the polynomials ( p(p(p(x))) )^n and so on.

Row sums are 2^(2*n); row sums of P^2 are 5^(2*n), row sums of P^3 are 26^(2*n). In general, the row sums of P^k, k = 0,1,2,..., are equal to A003095(k)^(2*n).

The signed version of this array ( (-1)^k*binomial(2*n,k) )n,k>=0 is a left-inverse for A034839.

A034839 * P = A080928. (End)

T(n, k) = GegenbauerC(m, -n, -1) where m = k if kPeter Luschny, May 08 2016

G.f.: 1/(1-x*(y+1)^2). - Vladimir Kruchinin, Nov 22 2020