A289794 -id:A289794 - cp's OEIS frontend

A292480 p-INVERT of the odd positive integers, where p(S) = 1 - S^2.

0, 1, 6, 20, 56, 160, 480, 1456, 4384, 13136, 39360, 118064, 354272, 1062928, 3188736, 9565936, 28697632, 86093264, 258280512, 774841520, 2324523104, 6973567888, 20920705152, 62762119792, 188286360736, 564859074896, 1694577214656, 5083731648560

Offset: 0

Clark Kimberling, Oct 02 2017

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).
In the following guide to p-INVERT sequences using s = (1,3,5,7,9,...) = A005408, in some cases t(1,3,5,7,9,...) is a shifted (or differently indexed) version of the cited sequence:
p(S) *********** t(1,3,5,7,9,...)
- S               A003946
- S^2             A292480
- S^3             (not yet in OEIS)
(1 - S)^2           (not yet in OEIS)
(1 - S)^3           (not yet in OEIS)
- S - S^2         A289786
+ S - S^2         A289484
- S - 2 S^2       A289785
- S - 3 S^2       A289786
- S - 4 S^2       A289787
- S - 5 S^2       A289788
- S - 6 S^2       A289789
- S - 7 S^2       A289790
+ S - 2 S^2       A289791
- S + S^2 - S^3   A289792
+ S - 3 S^2       A289793
- S - S^2 - S^3   A289794

			s = (1,3,5,7,9,...), S(x) = x + 3 x^2 + 5 x^3 + 7 x^4 + ...,
p(S(x)) = 1 - ( x + 3 x^2 + 5 x^3 + 7 x^4 + ...)^2,
1/p(S(x)) = 1 + x^2 + 6 x^3 + 20 x^4 + 56 x^5 + ...
T(x) = (-1 + 1/p(S(x)))/x = x + 6 x^2 + 20 x^3 + 56 x^4 + ...
t(s) = (0,1,2,20,56,...).

Clark Kimberling, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (4,-5,6)

Cf. A005408, A292479.

I:=[0,1,6,20]; [n le 4 select I[n] else 4*Self(n-1)- 5*Self(n-2)+6*Self(n-3): n in [1..30]]; // Vincenzo Librandi, Oct 03 2017

z = 60; s = x (x + 1)/(1 - x)^2; p = 1 - s^2;
Drop[CoefficientList[Series[s, {x, 0, z}], x], 1] (* A005408 *)
Drop[CoefficientList[Series[1/p, {x, 0, z}], x], 1]  (* A292480 *)
Join[{0}, LinearRecurrence[{4, -5, 6}, {1, 6, 20}, 30]] (* Vincenzo Librandi, Oct 03 2017 *)

G.f.: x*(1 + x)^2/((1 - 3*x)*(1 - x + 2*x^2)).

a(n) = 4*a(n-1) - 5*a(n-2) + 6*a(n-3) for n >= 5.

A027789 a(n) = 2(n+1)binomial(n+3,4).

Original entry on oeis.org

4, 30, 120, 350, 840, 1764, 3360, 5940, 9900, 15730, 24024, 35490, 50960, 71400, 97920, 131784, 174420, 227430, 292600, 371910, 467544, 581900, 717600, 877500, 1064700, 1282554, 1534680, 1824970, 2157600, 2537040, 2968064, 3455760, 4005540, 4623150, 5314680

Offset: 1

Thi Ngoc Dinh (via R. K. Guy)

Number of 8-subsequences of [ 1, n ] with just 3 contiguous pairs.

Also the number of 3-cycles in the n+3 tetrahedral graph. - Eric W. Weisstein, Jul 12 2017

Harvey P. Dale, Table of n, a(n) for n = 1..1000
Eric Weisstein's World of Mathematics, Graph Cycle.
Eric Weisstein's World of Mathematics, Tetrahedral Graph.
Index entries for linear recurrences with constant coefficients, signature (6,-15,20,-15,6,-1).

Cf. A006470, A289792 (4-cycles), A289793 (5-cycles), A289794 (6-cycles).

[2*(n+1)*Binomial(n+3,4): n in [1..40]]; // Vincenzo Librandi, Jul 13 2017

A027789:=n->2*(n+1)*binomial(n+3,4): seq(A027789(n), n=1..60); # Wesley Ivan Hurt, Oct 23 2017

Table[2 (n + 1) Binomial[n + 3, 4], {n, 40}] (* Harvey P. Dale, Jan 20 2015 *)
LinearRecurrence[{6, -15, 20, -15, 6, -1}, {4, 30, 120, 350, 840, 1764},40] (* Harvey P. Dale, Jan 20 2015 *)
Table[n (1 + n)^2 (2 + n) (3 + n)/12, {n, 20}] (* Eric W. Weisstein, Jul 12 2017 *)
CoefficientList[Series[(2 (2 + 3 x))/(-1 + x)^6, {x, 0, 20}], x] (* Eric W. Weisstein, Jul 12 2017 *)

for(n=1,50, print1(2*(n+1)*binomial(n+3,4), ", ")) \\ G. C. Greubel, Oct 22 2017

G.f.: 2*(2+3x)*x/(1-x)^6.
a(n) = 2*A006470(n).
a(n) = C(n+1, 2)*C(n+3, 3). - Zerinvary Lajos, May 10 2005, corrected by R. J. Mathar, Feb 13 2016
a(n) = 6*a(n-1) - 15*a(n-2) + 20*a(n-3) - 15*a(n-4) + 6*a(n-5) - a(n-6). - Harvey P. Dale, Jan 20 2015
a(n) = Sum_{k=1..n+1} Sum_{i=1..n+1} (n-i+1) * C(k+1,k-1). - Wesley Ivan Hurt, Sep 21 2017
From Amiram Eldar, Jan 28 2022: (Start)
Sum_{n>=1} 1/a(n) = 61/6 - Pi^2.
Sum_{n>=1} (-1)^(n+1)/a(n) = Pi^2/2 - 8*log(2) + 5/6. (End)

A289792 Number of 4-cycles in the n-tetrahedral graph.

Original entry on oeis.org

0, 0, 0, 0, 90, 540, 1995, 5775, 14280, 31500, 63630, 119790, 212850, 360360, 585585, 918645, 1397760, 2070600, 2995740, 4244220, 5901210, 8067780, 10862775, 14424795, 18914280, 24515700, 31439850, 39926250, 50245650, 62702640, 77638365, 95433345, 116510400

Offset: 1

Eric W. Weisstein, Jul 12 2017

Extended to a(1)-a(5) using the formula.

Eric Weisstein's World of Mathematics, Graph Cycle
Eric Weisstein's World of Mathematics, Tetrahedral Graph
Index entries for linear recurrences with constant coefficients, signature (7, -21, 35, -35, 21, -7, 1).

Cf. A027789 (3-cycles), A289793 (5-cycles), A289794 (6-cycles).

Table[Binomial[n - 1, 4] (210 - 41 n + 7 n^2)/2, {n, 20}]
LinearRecurrence[{7, -21, 35, -35, 21, -7, 1}, {0, 0, 0, 0, 90, 540, 1995}, 20]
CoefficientList[Series[-((15 x^4 (6 - 6 x + 7 x^2))/(-1 + x)^7), {x, 0, 20}], x]

a(n) = binomial(n - 1, 4) * (210 - 41*n + 7*n^2)/2.
a(n) = 7*a(n-1)-21*a(n-2)+35*a(n-3)-35*a(n-4)+21*a(n-5)-7*a(n-6)+a(n-7).
G.f.: (-15*x^5*(6 - 6*x + 7*x^2))/(-1 + x)^7.

A289793 Number of 5-cycles in the n-tetrahedral graph.

Original entry on oeis.org

0, 0, 0, 0, 312, 3024, 14868, 51744, 145152, 350784, 759528, 1511136, 2810808, 4948944, 8324316, 13470912, 21088704, 32078592, 47581776, 69023808, 98163576, 137147472, 188568996, 255534048, 341732160, 451513920, 589974840, 763045920, 977591160, 1241512272

Offset: 1

Eric W. Weisstein, Jul 12 2017

Extended to a(1)-a(5) using the formula.

Eric Weisstein's World of Mathematics, Graph Cycle
Eric Weisstein's World of Mathematics, Tetrahedral Graph
Index entries for linear recurrences with constant coefficients, signature (8, -28, 56, -70, 56, -28, 8, -1).

Cf. A027789 (3-cycles), A289792 (4-cycles), A289794 (6-cycles).

Table[6 Binomial[n, 5] (-78 + 21 n + n^2), {n, 20}]
LinearRecurrence[{8, -28, 56, -70, 56, -28, 8, -1}, {0, 0, 0, 0, 312, 3024, 14868, 51744}, 20]
CoefficientList[Series[-((12 x^4 (-26 - 44 x + 49 x^2))/(-1 + x)^8), {x, 0, 20}], x]

a(n) = 6*binomial(n, 5)*(-78 + 21*n + n^2).
a(n) = 8*a(n-1)-28*a(n-2)+56*a(n-3)-70*a(n-4)+56*a(n-5)-28*a(n-6)+8*a(n-7)-a(n-8).
G.f.: (-12*x^5*(-26 - 44*x + 49*x^2))/(-1 + x)^8.

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A292480 p-INVERT of the odd positive integers, where p(S) = 1 - S^2.

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A027789 a(n) = 2*(n+1)*binomial(n+3,4).

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A289792 Number of 4-cycles in the n-tetrahedral graph.

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A289793 Number of 5-cycles in the n-tetrahedral graph.

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A027789 a(n) = 2(n+1)binomial(n+3,4).