A080061 -id:A080061 - cp's OEIS frontend

A001891 Hit polynomials; convolution of natural numbers with Fibonacci numbers F(2), F(3), F(4), ....

0, 1, 4, 10, 21, 40, 72, 125, 212, 354, 585, 960, 1568, 2553, 4148, 6730, 10909, 17672, 28616, 46325, 74980, 121346, 196369, 317760, 514176, 831985, 1346212, 2178250, 3524517, 5702824, 9227400, 14930285, 24157748, 39088098, 63245913, 102334080, 165580064

Offset: 0

N. J. A. Sloane, Simon Plouffe

a(n) is the sum of the n-th row of the triangle in A119457 for n > 0. - Reinhard Zumkeller, May 20 2006
Convolution of odds (A005408) with Fibonacci numbers (A000045). - Graeme McRae, Jun 06 2006
Equals row sums of triangle A152203. - Gary W. Adamson, Nov 29 2008
Define a triangle by T(n,0) = n*(n+1)+1, T(n,n) = 1, and T(r,c) = T(r-1,c) + T(r-2,c-1). This triangle starts: 1; 3,1; 7,2,1; 13,5,2,1; 21,12,4,2,1; the sum of terms in row n is a(n+1). - J. M. Bergot, Apr 23 2013
a(n) = number of k-tuples (u(1), u(2), ..., u(k)) with 1 <= u(1) < u(2) < ... < u(k) <= n such that u(i) - u(i-1) <= 2 for i = 2,...,k. Changing the bound from 2 to 3, then 4, then 5, yields A356619, A356620, A356621. The patterns suggest that the limiting sequence as the bound increases is A000295. - Clark Kimberling, Aug 24 2022

J. Riordan, The enumeration of permutations with three-ply staircase restrictions, unpublished memorandum, Bell Telephone Laboratories, Murray Hill, NJ, Oct 1963. (See A001883)
N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Simon Plouffe, Approximations de séries génératrices et quelques conjectures, Dissertation, Université du Québec à Montréal, 1992; arXiv:0911.4975 [math.NT], 2009.
Simon Plouffe, 1031 Generating Functions, Appendix to Thesis, Montreal, 1992
N. J. A. Sloane, Annotated copy of Riordan's Three-Ply Staircase paper (unpublished memorandum, Bell Telephone Laboratories, Murray Hill, NJ, Oct 1963)
Index entries for linear recurrences with constant coefficients, signature (3,-2,-1,1).

Partial sums of A001911.
A diagonal of triangle in A080061.
Right-hand column 5 of triangle A011794.
Cf. A001883-A001890. A152203.
Cf. A000295, A356619, A356620, A356621.

List([0..40], n-> Fibonacci(n+5) -2*n-5); # G. C. Greubel, Jul 06 2019

[Fibonacci(n+5)-(5+2*n): n in [0..40]]; // Vincenzo Librandi, Jun 07 2013

LinearRecurrence[{3,-2,-1,1}, {0,1,4,10}, 40] (* Vladimir Joseph Stephan Orlovsky, Feb 16 2012 *)
Table[Fibonacci[n+5] -(2*n+5), {n,0,40}] (* G. C. Greubel, Jul 06 2019 *)
maxDiff = 2;
Map[Length[Select[Map[{#, Max[Differences[#]]} &,
  Drop[Subsets[Range[#]], # + 1]], #[[2]] <= maxDiff &]] &,
  Range[16]] (* Peter J. C. Moses, Aug 14 2022 *)

a(n)=([0,1,0,0; 0,0,1,0; 0,0,0,1; 1,-1,-2,3]^n*[0;1;4;10])[1,1] \\ Charles R Greathouse IV, Apr 08 2016

[fibonacci(n+5) -2*n-5 for n in (0..40)] # G. C. Greubel, Jul 06 2019

G.f.: x*(1+x)/((1-x-x^2)*(1-x)^2). - Simon Plouffe in his 1992 dissertation
a(n) = Fibonacci(n+5) - (5+2*n). - Wolfdieter Lang
a(n) = a(n-1) + a(n-2) + (2n+1); a(-x)=0. - Barry E. Williams, Mar 27 2000
a(n) = 3*a(n-1) - 2*a(n-2) - a(n-3) + a(n-4). - Sam Lachterman (slachterman(AT)fuse.net), Sep 22 2003
a(n) - a(n-1) = A101220(2,1,n). - Ross La Haye, May 31 2006
a(n) = (-3 + (2^(-1-n)*((1-sqrt(5))^n*(-11+5*sqrt(5)) + (1+sqrt(5))^n*(11+5*sqrt(5)))) / sqrt(5) - 2*(1+n)). - Colin Barker, Mar 11 2017

A058057 Triangle giving coefficients of ménage hit polynomials.

Original entry on oeis.org

1, 1, 0, 1, 1, 0, 1, 3, 1, 1, 1, 6, 6, 8, 3, 1, 10, 20, 38, 35, 16, 1, 15, 50, 134, 213, 211, 96, 1, 21, 105, 385, 915, 1479, 1459, 675, 1, 28, 196, 952, 3130, 7324, 11692, 11584, 5413, 1, 36, 336, 2100, 9090, 28764, 65784, 104364, 103605, 48800

Offset: 0

N. J. A. Sloane, Dec 02 2000

Triangle of coefficients of polynomials P(n; x) = Permanent(M), where M=[m(i,j)] is n X n matrix defined by m(i,j)=x if 0<=i-j<=1 else m(i,j)=1. - Vladeta Jovovic, Jan 23 2003

			1; 1,0; 1,1,0; 1,3,1,1; 1,6,6,8,3; ...

J. Riordan, An Introduction to Combinatorial Analysis, Wiley, 1958, p. 198.

T. D. Noe, Rows n = 0..100 of triangle, flattened

Diagonals give A000271, A000426, A000222, A000386, A000450, A058085, A058086.

Cf. A080018, A080061.

V := proc(n) local k; add( binomial(2*n-k,k)*(n-k)!*(x-1)^k, k=0..n); end; W := proc(r,s) coeff( V(r),x,s ); end; a := (n,k)->W(n,n-k);

max = 9; f[x_, y_] := Sum[n!*((x*y)^n/(1 + x*(y-1))^(2*n+1)), {n, 0, max}]; Flatten[ MapIndexed[ Take[#1, #2[[1]]] & , CoefficientList[ Series[f[x, y], {x, 0, max}, {y, 0, max}], {x, y}]]] (*Jean-François Alcover, Jun 29 2012, after Vladeta Jovovic *)

G.f.: Sum(n!*(x*y)^n/(1+x*(y-1))^(2*n+1),n=0..infinity). [Vladeta Jovovic, Dec 13 2009]

A001887 Number of permutations p of {1,2,...,n} such that p(i) - i < 0 or p(i) - i > 2 for all i.

Original entry on oeis.org

1, 0, 0, 0, 1, 5, 33, 236, 1918, 17440, 175649, 1942171, 23396353, 305055960, 4280721564, 64330087888, 1030831875953, 17545848553729, 316150872317105, 6012076099604308, 120330082937778554

Offset: 0

N. J. A. Sloane

nonn

Previous name was: Hit polynomials.

J. Riordan, The enumeration of permutations with three-ply staircase restrictions, unpublished memorandum, Bell Telephone Laboratories, Murray Hill, NJ, Oct 1963. (See A001883)
N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Alois P. Heinz, Table of n, a(n) for n = 0..300
E. R. Canfield, N. C. Wormald, Menage numbers, bijections and P-recursiveness, Discr. Math. 63 (1987) 117, table Section 7.
P. Flajolet and R. Sedgewick, Analytic Combinatorics, 2009; see page 373
V. Kotesovec, Non-attacking chess pieces, 6ed, 2013, p. 224.
N. J. A. Sloane, Annotated copy of Riordan's Three-Ply Staircase paper (unpublished memorandum, Bell Telephone Laboratories, Murray Hill, NJ, Oct 1963)
D. Zeilberger, Automatic Enumeration of Generalized Menage Numbers, arXiv preprint arXiv:1401.1089 [math.CO], 2014.

Cf. A000255, A000271, A001883-A001891, A075851, A075852.

First column of A080061.

nmax = 21;
gf = 1/(x^2-1)(x-Sum[n! (x(x-1)/(x^3-2x-1))^n + O[x]^nmax, {n, 0, nmax}]);
CoefficientList[gf, x] (* Jean-François Alcover, Aug 19 2018 *)

G.f.: (1/(x^2-1))*(x-Sum_{n>=0} n!*(x*(x-1)/(x^3-2*x-1))^n). - Vladeta Jovovic, Jun 30 2007
D-finite with recurrence (P. Flajolet, 1997): a(n) = (n-1)*a(n-1) + (n+2)*a(n-2) - (3*n-13)*a(n-3) - (2*n-8)*a(n-4) + (3*n-15)*a(n-5) + (n-4)*a(n-6) - (n-7)*a(n-7) - a(n-8), n>8.
a(n) ~ exp(-3) * n!. - Vaclav Kotesovec, Sep 10 2014

More terms from Vladimir Baltic and Vladeta Jovovic, Jan 05 2003

New name from Vaclav Kotesovec using a former comment by Vladimir Baltic and Vladeta Jovovic, Sep 16 2014

A001890 Hit polynomials.

Original entry on oeis.org

1, 10, 34, 206, 1351, 10543, 92708, 912884, 9917445, 117838808, 1519483258, 21128310078, 315093762147, 5016410089130, 84909414423784, 1522548805068310, 28830824064870329, 574880701875755325, 12039866150973004846, 264230694283295736788, 6063848537910027941323

Offset: 3

N. J. A. Sloane

nonn

J. Riordan, The enumeration of permutations with three-ply staircase restrictions, unpublished memorandum, Bell Telephone Laboratories, Murray Hill, NJ, Oct 1963. (See A001883)
N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

N. J. A. Sloane, Annotated copy of Riordan's Three-Ply Staircase paper (unpublished memorandum, Bell Telephone Laboratories, Murray Hill, NJ, Oct 1963)

Fourth column of A080061.

Cf. A001883-A001891.

(* Program not suitable to compute more than a dozen terms *)
M[n_] := Table[If[0 <= i - j <= 2, x, 1], {i, 1, n}, {j, 1, n}];
a[n_] := Coefficient[Permanent[M[n]], x, 3];
Table[an = a[n]; Print[n, " ", an]; an, {n, 3, 15}] (* Jean-François Alcover, Jan 12 2018 *)

More terms from Vladeta Jovovic, Vladimir Baltic, Jan 23 2003

a(21)-a(23) from Vaclav Kotesovec, Oct 10 2017

A001888 Hit polynomials.

Original entry on oeis.org

1, 1, 1, 4, 21, 122, 849, 6719, 59873, 593686, 6483027, 77309508, 999569969, 13927664613, 208044077633, 3316415176712, 56193522437465, 1008499234742158, 19110533105114433, 381290148293261075, 7989548583615385153, 175418232627066361514, 4027213499971769192903, 96489861550805864197512

Offset: 1

N. J. A. Sloane

nonn

J. Riordan, The enumeration of permutations with three-ply staircase restrictions, unpublished memorandum, Bell Telephone Laboratories, Murray Hill, NJ, Oct 1963. (See A001883)
N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

N. J. A. Sloane, Annotated copy of Riordan's Three-Ply Staircase paper (unpublished memorandum, Bell Telephone Laboratories, Murray Hill, NJ, Oct 1963)

Second column of A080061.

Cf. A001883-A001891.

More terms from Vladeta Jovovic, Vladimir Baltic, Jan 23 2003

a(21)-a(24) from Vaclav Kotesovec, Oct 10 2017

A001889 Hit polynomials.

Original entry on oeis.org

1, 4, 8, 38, 209, 1400, 10849, 95516, 938441, 10179149, 120748974, 1554688240, 21588485677, 321561688748, 5113709884902, 86469576394028, 1549114961438599, 29309613834899864, 583986501706555597, 12222118836543561940, 268060239251978779839, 6148136982942476816891

Offset: 2

N. J. A. Sloane

nonn

J. Riordan, The enumeration of permutations with three-ply staircase restrictions, unpublished memorandum, Bell Telephone Laboratories, Murray Hill, NJ, Oct 1963. (See A001883)
N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

N. J. A. Sloane, Annotated copy of Riordan's Three-Ply Staircase paper (unpublished memorandum, Bell Telephone Laboratories, Murray Hill, NJ, Oct 1963)

Third column of A080061.

Cf. A001883-A001891.

More terms from Vladeta Jovovic, Vladimir Baltic, Jan 23 2003

a(21)-a(23) from Vaclav Kotesovec, Oct 10 2017

cp's OEIS Frontend

A001891 Hit polynomials; convolution of natural numbers with Fibonacci numbers F(2), F(3), F(4), ....

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A058057 Triangle giving coefficients of ménage hit polynomials.

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A001887 Number of permutations p of {1,2,...,n} such that p(i) - i < 0 or p(i) - i > 2 for all i.

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A001890 Hit polynomials.

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