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A034870 Even-numbered rows of Pascal's triangle. 23
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, 1 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,3

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 0s 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

LINKS

Reinhard Zumkeller, Rows n=0..150 of triangle, flattened

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.

Index entries for triangles and arrays related to Pascal's triangle

FORMULA

a(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 the generalized Riordan array (1, (1 + x)^2).

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 k<n else 2*n-k. - Peter Luschny, May 08 2016

EXAMPLE

Triangle begins:

  1;

  1, 2,  1;

  1, 4,  6,  4,  1;

  1, 6, 15, 20, 15, 6, 1;

  ...

MAPLE

T := (n, k) -> simplify(GegenbauerC(`if`(k<n, k, 2*n-k), -n, -1));

for n from 0 to 6 do seq(T(n, k), k=0..2*n) od; # Peter Luschny, May 08 2016

MATHEMATICA

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

PROG

(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

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.

Sequence in context: A156579 A190284 A273891 * A264622 A275017 A141036

Adjacent sequences:  A034867 A034868 A034869 * A034871 A034872 A034873

KEYWORD

nonn,tabf,easy

AUTHOR

N. J. A. Sloane

STATUS

approved

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Last modified August 18 04:26 EDT 2018. Contains 313821 sequences. (Running on oeis4.)