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A027378
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Expansion of (1+x^2-x^3)/(1-x)^4.
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1
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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
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OFFSET
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0,2
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COMMENTS
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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
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LINKS
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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).
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FORMULA
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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
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MATHEMATICA
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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 *)
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PROG
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(Magma) [(n^3 +9*n^2 +8*n +6)/6: n in [0..50]]; // G. C. Greubel, Jul 30 2022
(SageMath) [(n^3 +9*n^2 +8*n +6)/6 for n in (0..50)] # G. C. Greubel, Jul 30 2022
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CROSSREFS
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Cf. A006416, A006503, A034857, A116721.
Appears to be first differences of A252814.
First differences at A027379 (omitting first term).
Sequence in context: A009907 A301159 A298023 * A131177 A092498 A301165
Adjacent sequences: A027375 A027376 A027377 * A027379 A027380 A027381
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KEYWORD
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nonn,easy
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AUTHOR
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N. J. A. Sloane
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STATUS
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approved
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