A096957 Fourth column (m=3) of (1,6)-Pascal triangle A096956.

6, 19, 40, 70, 110, 161, 224, 300, 390, 495, 616, 754, 910, 1085, 1280, 1496, 1734, 1995, 2280, 2590, 2926, 3289, 3680, 4100, 4550, 5031, 5544, 6090, 6670, 7285, 7936, 8624, 9350, 10115, 10920, 11766, 12654, 13585, 14560, 15580, 16646, 17759, 18920

Offset

0,1

Comments

If Y is a 6-subset of an n-set X then, for n>=8, a(n-8) is the number of 3-subsets of X having at most one element in common with Y. - Milan Janjic, Dec 16 2007

Links

Vincenzo Librandi, Table of n, a(n) for n = 0..3000

Index entries for linear recurrences with constant coefficients, signature (4,-6,4,-1).

Formula

a(n) = A096956(n+3, 3) = 6*b(n) - 5*b(n-1) = (n+18)*binomial(n+2, 2)/3, with b(n) = A000292(n) = binomial(n+3, 3).

G.f.: (6-5*x)/(1-x)^4.

a(n) = 4*a(n-1) - 6*a(n-2) + 4*a(n-3) - a(n-4) for n>3. - Vincenzo Librandi, Apr 19 2017

E.g.f.: exp(x)*(36 + 78*x + 24*x^2 + x^3)/6. - Stefano Spezia, May 02 2025

From Amiram Eldar, Oct 26 2025: (Start)

Sum_{n>=0} 1/a(n) = 166145857/555434880.

Sum_{n>=0} (-1)^n/a(n) = 12*log(2)/17 - 39889139/111086976. (End)

Mathematica

CoefficientList[Series[(6 - 5*x)/(1 - x)^4, {x, 0, 40}], x] (* Wesley Ivan Hurt, Apr 18 2017 *)
LinearRecurrence[{4, -6, 4, -1}, {6, 19, 40, 70}, 50] (* Vincenzo Librandi, Apr 19 2017 *)

Prog

(Magma) I:=[6, 19, 40, 70]; [n le 4 select I[n] else 4*Self(n-1)- 6*Self(n-2)+4*Self(n-3)-Self(n-4): n in [1..50]]; // Vincenzo Librandi, Apr 19 2017

Crossrefs

Cf. A000292, A096956.

Cf. other columns: A096958 (m = 4), A096959 (m = 5), A097297 (m = 6), A097298 (m = 7), A097299 (m = 8), A097300 (m = 9).

Sequence in context: A106398 A179986 A054567 * A272811 A273206 A273394

Adjacent sequences: A096954 A096955 A096956 * A096958 A096959 A096960

Keyword

nonn,easy

Author

Wolfdieter Lang, Aug 13 2004

Status

approved