A027378 Expansion of (1+x^2-x^3)/(1-x)^4.

1, 4, 11, 23, 41, 66, 99, 141, 193, 256, 331, 419, 521, 638, 771, 921, 1089, 1276, 1483, 1711, 1961, 2234, 2531, 2853, 3201, 3576, 3979, 4411, 4873, 5366, 5891, 6449, 7041, 7668, 8331, 9031, 9769, 10546

Offset: 0

Author

N. J. A. Sloane

Keywords

• nonn
• easy

Comments

If Y is a 3-subset of an n-set X then, for n>=4, a(n-4) is the number of (n-3)-subsets of X which do not have exactly one element in common with Y. - Milan Janjic, Dec 28 2007

Links

• G. C. Greubel, Table of n, a(n) for n = 0..1000
• Index entries for linear recurrences with constant coefficients, signature (4,-6,4,-1).

Crossrefs

Cf. A006416, A006503, A034857, A116721.

Appears to be first differences of A252814.

First differences at A027379 (omitting first term).

Programs

Magma

[(n^3 +9*n^2 +8*n +6)/6: n in [0..50]]; // G. C. Greubel, Jul 30 2022

Mathematica

CoefficientList[Series[(1+x^2-x^3)/(1-x)^4,{x,0,50}],x] (* or *) LinearRecurrence[{4,-6,4,-1},{1,4,11,23},50] (* Harvey P. Dale, May 17 2021 *)

SageMath

[(n^3 +9*n^2 +8*n +6)/6 for n in (0..50)] # G. C. Greubel, Jul 30 2022

Formula

a(n) = binomial(n+4, 3) - 3*(n+1). - Milan Janjic, Dec 28 2007 [Correction by Mathew Englander, Feb 03 2022]

a(n) = A006503(n) + 1 = A034857(n) + 5 = A116721(n+2) - 1 = A006416(n+1) + 3. - Mathew Englander, Feb 03 2022

E.g.f.: (1/6)*(6 + 18*x + 12*x^2 + x^3)*exp(x). - G. C. Greubel, Jul 30 2022