A289780 - OEIS

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A289780

p-INVERT of the positive integers (A000027), where p(S) = 1 - S - S^2.

1, 4, 14, 47, 156, 517, 1714, 5684, 18851, 62520, 207349, 687676, 2280686, 7563923, 25085844, 83197513, 275925586, 915110636, 3034975799, 10065534960, 33382471801, 110713382644, 367182309614, 1217764693607, 4038731742156, 13394504020957, 44423039068114 (list; graph; refs; listen; history; text; internal format)

	OFFSET	`0,2`
	COMMENTS	`Suppose s = (c(0), c(1), c(2), ...) is a sequence and p(S) is a polynomial. Let S(x) = c(0)x + c(1)x^2 + c(2)x^3 + ... and T(x) = (-p(0) + 1/p(S(x)))/x. The p-INVERT of s is the sequence t(s) of coefficients in the Maclaurin series for T(x).` `Taking p(S) = 1 - S gives the INVERT transform of s, so that p-INVERT is a generalization of the INVERT transform (e.g., A033453).` `Guide to p-INVERT sequences using p(S) = 1 - S - S^2:` `t(A000012) = t(1,1,1,1,1,1,1,...) = A001906` `t(A000290) = t(1,4,9,16,25,36,...) = A289779` `t(A000027) = t(1,2,3,4,5,6,7,8,...) = A289780` `t(A000045) = t(1,2,3,5,8,13,21,...) = A289781` `t(A000032) = t(2,1,3,4,7,11,14,...) = A289782` `t(A000244) = t(1,3,9,27,81,243,...) = A289783` `t(A000302) = t(1,4,16,64,256,...) = A289784` `t(A000351) = t(1,5,25,125,625,...) = A289785` `t(A005408) = t(1,3,5,7,9,11,13,...) = A289786` `t(A005843) = t(2,4,6,8,10,12,14,...) = A289787` `t(A016777) = t(1,4,7,10,13,16,...) = A289789` `t(A016789) = t(2,5,8,11,14,17,...) = A289790` `t(A008585) = t(3,6,9,12,15,18,...) = A289795` `t(A000217) = t(1,3,6,10,15,21,...) = A289797` `t(A000225) = t(1,3,7,15,31,63,...) = A289798` `t(A000578) = t(1,8,27,64,625,...) = A289799` `t(A000984) = t(1,2,6,20,70,252,...) = A289800` `t(A000292) = t(1,4,10,20,35,56,...) = A289801` `t(A002620) = t(1,2,4,6,9,12,16,...) = A289802` `t(A001906) = t(1,3,8,21,55,144,...) = A289803` `t(A001519) = t(1,1,2,5,13,34,...) = A289804` `t(A103889) = t(2,1,4,3,6,5,8,7,,...) = A289805` `t(A008619) = t(1,1,2,2,3,3,4,4,...) = A289806` `t(A080513) = t(1,2,2,3,3,4,4,5,...) = A289807` `t(A133622) = t(1,2,1,3,1,4,1,5,...) = A289809` `t(A000108) = t(1,1,2,5,14,42,...) = A081696` `t(A081696) = t(1,1,3,9,29,97,...) = A289810` `t(A027656) = t(1,0,2,0,3,0,4,0,5...) = A289843` `t(A175676) = t(1,0,0,2,0,0,3,0,...) = A289844` `t(A079977) = t(1,0,1,0,2,0,3,...) = A289845` `t(A059841) = t(1,0,1,0,1,0,1,...) = A289846` `t(A000040) = t(2,3,5,7,11,13,...) = A289847` `t(A008578) = t(1,2,3,5,7,11,13,...) = A289828` `t(A000142) = t(1!, 2!, 3!, 4!, ...) = A289924` `t(A000201) = t(1,3,4,6,8,9,11,...) = A289925` `t(A001950) = t(2,5,7,10,13,15,...) = A289926` `t(A014217) = t(1,2,4,6,11,17,29,...) = A289927` `t(A000045) = t(0,1,1,2,3,5,...) = A289975 (* indicates prepended 0's)` `t(A000045) = t(0,0,1,1,2,3,5,...) = A289976` `t(A000045) = t(0,0,0,1,1,2,3,5,...) = A289977` `t(A290990) = t(0,1,2,3,4,5,...) = A290990` `t(A290990) = t(0,0,1,2,3,4,5,...) = A290991` `t(A290990*) = t(0,0,01,2,3,4,5,...) = A290992`
	LINKS	`Clark Kimberling, Table of n, a(n) for n = 0..1000` `Index entries for linear recurrences with constant coefficients, signature (5, -7, 5, -1)`
	FORMULA	`G.f.: (1 - x + x^2)/(1 - 5 x + 7 x^2 - 5 x^3 + x^4).` `a(n) = 5a(n-1) - 7a(n-2) + 5*a(n-3) - a(n-4).`
	EXAMPLE	`Example 1: s = (1,2,3,4,5,6,...) = A000027 and p(S) = 1 - S.` `S(x) = x + 2x^2 + 3x^3 + 4x^4 + ...` `p(S(x)) = 1 - (x + 2x^2 + 3x^3 + 4x^4 + ... )` `- p(0) + 1/p(S(x)) = -1 + 1 + x + 3x^2 + 8x^3 + 21x^4 + ...` `T(x) = 1 + 3x + 8x^2 + 21x^3 + ...` `t(s) = (1,3,8,21,...) = A001906.` `***` `Example 2: s = (1,2,3,4,5,6,...) = A000027 and p(S) = 1 - S - S^2.` `S(x) = x + 2x^2 + 3x^3 + 4x^4 + ...` `p(S(x)) = 1 - ( x + 2x^2 + 3x^3 + 4x^4 + ...) - ( x + 2x^2 + 3x^3 + 4x^4 + ...)^2` `- p(0) + 1/p(S(x)) = -1 + 1 + x + 4x^2 + 14x^3 + 47x^4 + ...` `T(x) = 1 + 4x + 14x^2 + 47x^3 + ...` `t(s) = (1,4,14,47,...) = A289780.`
	MATHEMATICA	`z = 60; s = x/(1 - x)^2; p = 1 - s - s^2;` `Drop[CoefficientList[Series[s, {x, 0, z}], x], 1] (* A000027 )` `Drop[CoefficientList[Series[1/p, {x, 0, z}], x], 1] ( A289780 *)`
	PROG	`(PARI) x='x+O('x^99); Vec((1-x+x^2)/(1-5x+7x^2-5x^3+x^4)) \\ Altug Alkan, Aug 13 2017` `(GAP)` `P:=[1, 4, 14, 47];; for n in [5..10^2] do P[n]:=5P[n-1]-7P[n-2]+5P[n-3]-P[n-4]; od; P; # Muniru A Asiru, Sep 03 2017`
	CROSSREFS	`Cf. A000027.` `Sequence in context: A094789 A273714 A082574 * A320404 A137284 A228178` `Adjacent sequences: A289777 A289778 A289779 * A289781 A289782 A289783`
	KEYWORD	`nonn,easy`
	AUTHOR	`Clark Kimberling, Aug 10 2017`
	STATUS	`approved`

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Last modified January 24 20:36 EST 2021. Contains 340411 sequences. (Running on oeis4.)