R. K. Guy - cp's OEIS frontend

A259472 Coefficients in an asymptotic expansion of A003319(n)/n! in falling factorials.

1, -2, -1, -4, -19, -110, -745, -5752, -49775, -476994, -5016069, -57462828, -712732987, -9521244982, -136356161873, -2084860795232, -33907076207495, -584602069590058, -10652917092110429, -204604743619641620, -4131502481607654739, -87507494737954740126

Offset: 0

N. J. A. Sloane, Jul 03 2015, following a suggestion from R. K. Guy, Apr 29 1974

sign

			A003319(n) / n! ~ 1 - 2/n - 1/(n*(n-1)) - 4/(n*(n-1)*(n-2)) - 19/(n*(n-1)*(n-2)*(n-3)) - 110/(n*(n-1)*(n-2)*(n-3)*(n-4)) - 745/(n*(n-1)*(n-2)*(n-3)*(n-4)*(n-5)) -  ... [coefficients are A259472]
A003319(n) / n! ~ 1 - 2/n - 1/n^2 - 5/n^3 - 32/n^4 - 253/n^5 - 2381/n^6 - ... [coefficients are A260503]

Vaclav Kotesovec, Table of n, a(n) for n = 0..446
L. Comtet, Sur les coefficients de l'inverse de la série formelle Sum n! t^n, Comptes Rend. Acad. Sci. Paris, A 275 (1972), 569-572.
L. Comtet, Series inversions, C. R. Acad. Sc. Paris, t. 275 (25 septembre 1972), 569-572. (Annotated scanned copy)
R. K. Guy, Letter to N. J. A. Sloane, Mar 1974

Cf. A003319, A260503, A261214, A261239, A261253, A261254, A059439.

CoefficientList[Series[1/Sum[k! * x^k, {k, 0, 20}]^2, {x, 0, 20}], x] (* Vaclav Kotesovec, Aug 03 2015 *)
CoefficientList[Assuming[Element[x,Reals], Series[E^(2/x) * x^2 / ExpIntegralEi[1/x]^2, {x,0,25}]], x] (* Vaclav Kotesovec, Aug 03 2015 *)

From Vaclav Kotesovec, Aug 12 2015: (Start)
G.f.: (1/Sum(k! x^k))^2.
Expansion of (1-g(x))^2, where g(x) is the g.f. of A003319.
a(n) ~ -2*n! * (1 - 3/n - 4/n^3 - 33/n^4 - 283/n^5 - 2785/n^6 - 31291/n^7 - 395360/n^8 - 5544754/n^9 - 85427259/n^10), for coefficients see A261214.
For n>0, a(n) = A059439(n) - 2*A003319(n).
For n>0, a(n) = Sum_{k=1..n} A260503(k) * Stirling1(n-1, k-1).
(End)

More terms from Vaclav Kotesovec, Aug 01 2015
New name from Vaclav Kotesovec, Aug 12 2015
Entry revised by Vaclav Kotesovec, Aug 12 2015

A242497 Sides of (Heronian) triangles where sides are consecutive integers and area is an integer.

Original entry on oeis.org

3, 4, 5, 13, 14, 15, 51, 52, 53, 193, 194, 195, 723, 724, 725, 2701, 2702, 2703, 10083, 10084, 10085, 37633, 37634, 37635, 140451, 140452, 140453, 524173, 524174, 524175, 1956243, 1956244, 1956245, 7300801, 7300802, 7300803, 27246963, 27246964, 27246965

Offset: 1

R. K. Guy and Charles R Greathouse IV, May 16 2014

Let the edge lengths of the triangle be 2x-1, 2x, 2x+1 so that area = sqrt{3x * x * (x-1) * (x+1)} and we need x^2 - 1 to be of shape 3y^2. That is, x/y is an even rank convergent to the continued fraction of sqrt(3) and x is A001075.

The intermediate length sides are given by A003500(n), n >= 1. Note that A003500(0) = 2 corresponds to the degenerate (Heronian) triangle with sides {1, 2, 3} and area 0. - Daniel Forgues, May 28 2014

Nakane Genkei (Nakane the Elder), Shichijo Beki Yenshiki, 1691.

Harvey P. Dale, Table of n, a(n) for n = 1..1000
David Eugene Smith and Yoshio Mikami, A history of Japanese mathematics, Dover, 2004, p. 168.
Index entries for linear recurrences with constant coefficients, signature (-1,-1,4,4,4,-1,-1,-1).

A016064 is the main entry for this sequence.

LinearRecurrence[{-1,-1,4,4,4,-1,-1,-1},{3,4,5,13,14,15,51,52},40] (* Harvey P. Dale, May 04 2021 *)

Vec((-3*x^7 - 5*x^6 - 6*x^5 + 4*x^4 + 10*x^3 + 12*x^2 + 7*x + 3)/(x^8 + x^7+ x^6 - 4*x^5 - 4*x^4 - 4*x^3 + x^2 + x + 1)+O(x^99))

G.f.: (-3*x^7 - 5*x^6 - 6*x^5 + 4*x^4 + 10*x^3 + 12*x^2 + 7*x + 3)/ ((1+x+x^2)*(1-4*x^3+x^6)). - R. J. Mathar, May 30 2023

A214525 a(n) = 7a(n-1) - 23a(n-2) + 49a(n-3) - 49a(n-4) with a(0)=0, a(1)=1, a(2)=7, a(3)=19.

Original entry on oeis.org

0, 1, 7, 19, 21, 4, 133, 937, 2667, 3439, 2128, 20569, 132867, 392743, 596869, 647596, 3539109, 19881229, 60254719, 106198903, 158297664, 643809889, 3117087967, 9564827611, 19050869061, 34555674196, 119658973525, 507648339217, 1561117435059, 3421971910543

Offset: 0

N. J. A. Sloane, Aug 07 2012, based on a posting to the Sequence Fans Mailing List by R. K. Guy, Jul 29 2009

This is a divisibility sequence.

It factors over the Eisenstein-Jacobi integers into two 2nd order sequences (with w^3 = 1): 0, 1, w+3, 3w+5, 4w+5, 2, -12w-1, -29w+3, ... and its conjugate (replace w by w^2). The relation for this is a(n) = (w+3)a(n-1) - (2w+3)a(n-2).

Bruno Berselli, Table of n, a(n) for n = 0..1000
R. K. Guy, New sequence?, SeqFan, 2009
Hugh Williams and R. K. Guy, Some fourth-order linear divisibility sequences, Intl. J. Number Theory, 7(5) (2011), 1255-1277.
Index entries for linear recurrences with constant coefficients, signature (7,-23,49,-49).

RecurrenceTable[{a[0] == 0, a[1] == 1, a[2] == 7, a[3] == 19, a[n] == 7 a[n - 1] - 23 a[n - 2] + 49 a[n - 3] - 49 a[n - 4]}, a[n], {n, 0, 29}] (* Bruno Berselli, Aug 08 2012 *)
LinearRecurrence[{7,-23,49,-49},{0,1,7,19},30] (* Harvey P. Dale, Jan 02 2023 *)

G.f.: x*(1-7*x^2) /(1-7*x+23*x^2-49*x^3+49*x^4). - Bruno Berselli, Aug 08 2012

a(9) corrected by Bruno Berselli, Aug 08 2012

A212804 Expansion of (1 - x)/(1 - x - x^2).

Original entry on oeis.org

1, 0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, 377, 610, 987, 1597, 2584, 4181, 6765, 10946, 17711, 28657, 46368, 75025, 121393, 196418, 317811, 514229, 832040, 1346269, 2178309, 3524578, 5702887, 9227465, 14930352, 24157817, 39088169, 63245986, 102334155, 165580141, 267914296, 433494437, 701408733, 1134903170, 1836311903, 2971215073, 4807526976

Offset: 0

N. J. A. Sloane, May 27 2012, following a suggestion from R. K. Guy

A variant of the Fibonacci number A000045.
Number of compositions of n into parts >= 2. - Joerg Arndt, Aug 13 2012
From Petros Hadjicostas, Jan 08 2019: (Start)
For n >= 0, a(n) is the number of unmarked circular binary words (necklaces) of length n+1 with exactly one occurrence of the pattern 00 (provided we allow the strings of length 1, i.e., 0 and 1, to wrap around themselves on a circle to form strings of length 2). See the comments for array A320341.
Using MacMahon's bijection between necklaces and cyclic compositions, we conclude that a(n) is also the number of (unmarked) cyclic compositions of n+1 with exactly one 1.
Removing the single 1 from each cyclic composition of n+1, we get all linear compositions of n with each part >= 2, which is what is stated above by Joerg Arndt. (End)

			From _Petros Hadjicostas_, Jan 08 2019: (Start)
For n = 6, we have a(6) = 5. The binary necklaces of length n+1 = 7 with exactly one occurrence of 00 are as follows: 0011111, 0010111, 0011011, 0011101, and 0010101.
The corresponding cyclic compositions of n+1 = 7 with exactly one 1 (under MacMahon's bijection) are as follows: 1+6, 1+2+4, 1+3+3, 1+4+2, 1+2+2+2.
Of course, removing the 1 from the cyclic composition, we get a (linear) composition of n = 6 with parts >= 2 (as stated above by _Joerg Arndt_): 6, 2+4, 3+3, 4+2, 2+2+2. (For linear compositions, 2+4 is not the same as 4+2.) (End)

John H. Conway and Richard K. Guy, The Book of Numbers, New York: Springer-Verlag, 1996. See p. 84.

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Jean-Luc Baril, Sergey Kirgizov, and Armen Petrossian, Dyck Paths with catastrophes modulo the positions of a given pattern, Australasian J. Comb. (2022) Vol. 84, No. 2, 398-418.
Jia Huang, Partially Palindromic Compositions, J. Int. Seq. (2023) Vol. 26, Art. 23.4.1. See pp. 4, 11.
J. J. Madden, A generating function for the distribution of runs in binary words, arXiv:1707.04351 [math.CO], 2017, Theorem 1.1, r=1 and k=0.
Index entries for linear recurrences with constant coefficients, signature (1,1).

Cf. A000045, A105809 (alternating row sums).

Column k=1 of A320341.

[Fibonacci(n + 1) - Fibonacci(n): n in [0..50]]; // Vincenzo Librandi, Dec 09 2012

a := n -> -I^n*ChebyshevU(n-2, -I/2):
seq(simplify(a(n)), n = 0..49);  # Peter Luschny, Dec 03 2023

Table[Fibonacci[n-1], {n, 0, 40}] (* Vladimir Reshetnikov, Sep 24 2016 *)

def A212804():
    a, b = True, False
    x, y = 1, 0
    while True:
        yield x if a else -x
        x, y = y, x - y
        a, b = b, a
a = A212804()
print([next(a) for  in range(50)]) # _Peter Luschny, Mar 19 2020

G.f.: 1/(1 - (Sum_{k >= 2} x^k)). - Joerg Arndt, Aug 13 2012
a(n) = Fibonacci(n+1) - Fibonacci(n). - Arkadiusz Wesolowski, Oct 29 2012
G.f.: 1 - x*Q(0) where Q(k) = 1 - (1 + x)/(1 - x/(x - 1/Q(k+1))); (continued fraction). - Sergei N. Gladkovskii, Mar 06 2013
G.f.: 3*x^3/(3*x - Q(0)) - x^2 + 1, where Q(k) = 1 - 1/(4^k - x*16^k/(x*4^k - 1/(1 + 1/(2*4^k - 4*x*16^k/(2*x*4^k + 1/Q(k+1)))))); (continued fraction). - Sergei N. Gladkovskii, May 21 2013
G.f.: G(0)*(1 - x)/(2 - x), where G(k) = 1 + 1/(1 - (x*(5*k - 1))/((x*(5*k + 4)) - 2/G(k+1))); (continued fraction). - Sergei N. Gladkovskii, Jun 15 2013
G.f.: 1 + Q(0)*x^2/2, where Q(k) = 1 + 1/(1 - x*(2*k + 1 + x)/( x*(2*k + 2 + x) + 1/Q(k+1) )); (continued fraction). - Sergei N. Gladkovskii, Aug 29 2013
a(n) = Sum_{k=0..n} (C(k, n-k) - C(k, n-k-1)). - Peter Luschny, Oct 01 2014
a(n) = (2^(-1-n)*((1 - sqrt(5))^n*(1 + sqrt(5)) + (-1 + sqrt(5))*(1 + sqrt(5))^n))/sqrt(5). - Colin Barker, Sep 25 2016
a(n) = A000045(n-1), n >= 1. - R. J. Mathar, Apr 14 2018
E.g.f.: exp((1 - sqrt(5))*x/2)*(3 + sqrt(5) + 2*exp(sqrt(5)*x))/(5 + sqrt(5)). - Stefano Spezia, Mar 09 2025

A171099 a(n) = number of solutions (x,y) (with 0 <= x <= y) to x(x+1)/2 + y(y+1)/2 = n!.

Original entry on oeis.org

1, 1, 1, 2, 1, 2, 0, 1, 1, 1, 2, 0, 0, 1, 0, 2, 1, 2, 0, 0, 0, 2, 0, 0, 2, 0, 0, 8, 1, 2, 0, 0, 4, 4, 16, 0, 0, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 16, 0, 0, 0, 0, 1, 0, 0, 0, 0, 32, 0, 0, 0, 0, 0, 0, 2, 0, 16, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 16, 2, 4, 0, 0, 0, 2, 1, 0, 0, 0, 0, 0, 0

Offset: 0

N. J. A. Sloane, Sep 24 2010, based on a posting by R. K. Guy to the Sequence Fans Mailing List, Sep 10 2010

nonn

			Initial solutions: (x,y,n) = (0,1,0), (0,1,1), (1,1,2), (0,3,3), (2,2,3), (2,6,4), (0,15,5), (5,14,5), (45,89,7), (89,269,8), (210,825,9), (760,2610,10), (1770,2030,10), none for n = 11 or 12, one for n = 13 (71504,85680,13) (found by _Ed Pegg Jr_), etc.

Cf. A000161, A152089 (n for which no solutions exist), A180590 (n for which solutions exist).

a(n) = A000161(8*n! + 2). - Max Alekseyev, Dec 12 2011

Corrected and extended (with data from Georgi Guninski, at the suggestion of N. J. A. Sloane) by D. S. McNeil, Sep 26 2010

A171064 G.f.: -x(x-1)(1+x)/(1-x-7*x^2-x^3+x^4).

Original entry on oeis.org

0, 1, 1, 7, 15, 64, 175, 631, 1905, 6433, 20224, 66529, 212625, 692119, 2226799, 7217728, 23284815, 75343591, 243328225, 786800449, 2542156800, 8217744577, 26556314401, 85835882791, 277405671375, 896595420736, 2897714688751

Offset: 0

R. J. Mathar, at the request of R. K. Guy, Sep 03 2010

The member k=7 of a family of sequences starting 0,1,1,k with recurrence a(n) = a(n-1)+k*a(n-2)+a(n-3)-a(n-4).

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Hugh Williams, R. K. Guy, Some fourth-order linear divisibility sequences, Intl. J. Number Theory 7 (5) (2011) 1255-1277
Index entries for linear recurrences with constant coefficients, signature (1,7,1,-1).

Cf. A116201 (k=1), A105309 (k=2), A152090 (k=3), A007598 (k=4), A005178 (k=5), A003757 (k=6).

I:=[0, 1, 1, 7]; [n le 4 select I[n] else Self(n-1) + 7*Self(n-2) + Self(n-3) - Self(n-4): n in [1..30]]; // Vincenzo Librandi, Dec 19 2012

CoefficientList[Series[-x*(x - 1)*(1 + x)/(1 - x - 7*x^2 - x^3 + x^4), {x, 0, 40}], x] (* Vincenzo Librandi, Dec 19 2012 *)
LinearRecurrence[{1,7,1,-1},{0,1,1,7},30] (* Harvey P. Dale, Nov 15 2020 *)

a(n) = +a(n-1) +7*a(n-2) +a(n-3) -a(n-4).

The roots (r1..r4) of the characteristic polynomials for this "family" of sequences have the following form (not simplified) for k= 1,2,3,4,5,6.... r1=(sqrt(4*k+10+2*sqrt(4*k+9))+sqrt(4*k-6+2*sqrt(4*k+9)))/4. r2=(sqrt(4*k+10+2*sqrt(4*k+9))-sqrt(4*k-6+2*sqrt(4*k+9)))/4. r3=(-sqrt(4*k+10-2*sqrt(4*k+9))-sqrt(4*k-6-2*sqrt(4*k+9)))/4. r4=(-sqrt(4*k+10-2*sqrt(4*k+9))+sqrt(4*k-6-2*sqrt(4*k+9)))/4. For k=1,2,3, r3 and r4 are complex . Closed-form (not simplified) is as follows for all k (note:for k1-k3 set r3 and r4 =0 and round a(n) to nearest integer): a(n)=sqrt(4*k+9)/(4*k+9)*(((r1)^n+(r2)^n)-((r3)^n+(r4)^n)). [Tim Monahan, Sep 17 2011]

A171065 G.f. -x(x-1)(1+x)/(1-x-8*x^2-x^3+x^4).

Original entry on oeis.org

0, 1, 1, 8, 17, 81, 224, 881, 2737, 9928, 32481, 113761, 380800, 1313441, 4441121, 15215688, 51677297, 176530481, 600723424, 2049428881, 6980069457, 23799693448, 81088954561, 276417142721, 941948403200, 3210574806081

Offset: 0

R. J. Mathar, at the request of R. K. Guy, Sep 03 2010

The member k=8 of a family of sequences starting 0,1,1,k with recurrence a(n) = a(n-1)+k*a(n-2)+a(n-3)-a(n-4).

This is the case P1 = 1, P2 = -10, Q = 1 of the 3 parameter family of 4th-order linear divisibility sequences found by Williams and Guy. - Peter Bala, Mar 31 2014

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Hugh Williams, R. K. Guy, Some fourth-order linear divisibility sequences, Intl. J. Number Theory vol. 7 (5) (2011) 1255-1277
H. C. Williams and R. K. Guy, Some Monoapparitic Fourth Order Linear Divisibility Sequences Integers, Volume 12A (2012) The John Selfridge Memorial Volume
Index entries for linear recurrences with constant coefficients, signature (1,8,1,-1).

Cf. A116201 (k=1), A105309 (k=2), A152090 (k=3), A007598 (k=4), A005178 (k=5), A003757 (k=6). A100047.

I:=[0, 1, 1, 8]; [n le 4 select I[n] else Self(n-1) + 8*Self(n-2) + Self(n-3) - Self(n-4): n in [1..30]]; // Vincenzo Librandi, Dec 19 2012

CoefficientList[Series[-x*(x - 1)*(1 + x)/(1 - x - 8*x^2 - x^3 + x^4), {x, 0, 40}], x] (* Vincenzo Librandi, Dec 19 2012 *)
LinearRecurrence[{1,8,1,-1},{0,1,1,8},30] (* Harvey P. Dale, Dec 27 2017 *)

a(n)= +a(n-1) +8*a(n-2) +a(n-3) -a(n-4).
From Peter Bala, Mar 31 2014: (Start)
a(n) = ( T(n,alpha) - T(n,beta) )/(alpha - beta), where alpha = (1 + sqrt(41))/4 and beta = (1 - sqrt(41))/4 and T(n,x) denotes the Chebyshev polynomial of the first kind.
a(n) = the bottom left entry of the 2 X 2 matrix T(n, M), where M is the 2 X 2 matrix [0, 5/2; 1, 1/2].
a(n) = U(n-1,i*(1 + sqrt(2))/2)*U(n-1,i*(1 + sqrt(2))/2), where U(n,x) denotes the Chebyshev polynomial of the second kind.
See the remarks in A100047 for the general connection between Chebyshev polynomials and 4th-order linear divisibility sequences. (End)

A171066 G.f. -x(x-1)(1+x)/(1-x-9*x^2-x^3+x^4).

Original entry on oeis.org

0, 1, 1, 9, 19, 100, 279, 1189, 3781, 14661, 49600, 184141, 641421, 2333629, 8240959, 29700900, 105561739, 378777169, 1350292761, 4835148121, 17260998400, 61748847081, 220582688041, 788748162049, 2818480203099, 10076047502500

Offset: 0

R. J. Mathar, at the request of R. K. Guy, Sep 03 2010

The member k=9 of a family of sequences starting 0,1,1,k with recurrence a(n) = a(n-1)+k*a(n-2)+a(n-3)-a(n-4).

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Hugh Williams, R. K. Guy, Some fourth-order linear divisibility sequences, Intl. J. Number Theory vol. 7 (5) (2011) 1255-1277
Index entries for linear recurrences with constant coefficients, signature (1,9,1,-1).

Cf. A116201 (k=1), A105309 (k=2), A152090 (k=3), A007598 (k=4), A005178 (k=5), A003757 (k=6).

I:=[0, 1, 1, 9]; [n le 4 select I[n] else Self(n-1) + 9*Self(n-2) + Self(n-3) - Self(n-4): n in [1..30]]; // Vincenzo Librandi, Dec 19 2012

CoefficientList[Series[-x*(x - 1)*(1 + x)/(1 - x - 9*x^2 - x^3 + x^4), {x, 0, 40}], x] (* Vincenzo Librandi, Dec 19 2012 *)

a(n)= +a(n-1) +9*a(n-2) +a(n-3) -a(n-4)

A171067 G.f. -x(x-1)(1+x)/((x^2+3x+1)(x^2-4*x+1)).

Original entry on oeis.org

0, 1, 1, 10, 21, 121, 340, 1561, 5061, 20890, 72721, 285121, 1028160, 3931201, 14425201, 54480250, 201635301, 756931801, 2813339860, 10529812921, 39218508021, 146573045290, 546474598561, 2040893746561, 7612994269440

Offset: 0

R. J. Mathar, at the request of R. K. Guy, Sep 03 2010

The member k=10 of a family of sequences starting 0,1,1,k with recurrence a(n) = a(n-1)+k*a(n-2)+a(n-3)-a(n-4).

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Hugh Williams, R. K. Guy, Some fourth-order linear divisibility sequences, Intl. J. Number Theory vol. 7 (5) (2011) 1255-1277
Index entries for linear recurrences with constant coefficients, signature (1,10,1,-1).

Cf. A116201 (k=1), A105309 (k=2), A152090 (k=3), A007598 (k=4), A005178 (k=5), A003757 (k=6).

I:=[0, 1, 1, 10]; [n le 4 select I[n] else Self(n-1) + 10*Self(n-2) + Self(n-3) - Self(n-4): n in [1..30]]; // Vincenzo Librandi, Dec 19 2012

CoefficientList[Series[-x*(x - 1)*(1 + x)/((x^2 + 3*x + 1)*(x^2 - 4*x + 1)), {x, 0, 40}], x] (* Vincenzo Librandi, Dec 19 2012 *)
LinearRecurrence[{1,10,1,-1},{0,1,1,10},30] (* Harvey P. Dale, Dec 24 2017 *)

a(n)= +a(n-1) +10*a(n-2) +a(n-3) -a(n-4).

a(n)= -(-1)^n*A005248(n)/7 + 2*A001075(n)/7.

A171068 G.f. -x(x-1)(1+x)/(1-x-11*x^2-x^3+x^4).

Original entry on oeis.org

0, 1, 1, 11, 23, 144, 407, 2003, 6601, 28897, 103104, 425569, 1582009, 6337475, 24062039, 94930704, 364368599, 1426330907, 5505254161, 21464332033, 83084090112, 323270665729, 1253154734833, 4870751815931, 18895640474711

Offset: 0

R. J. Mathar, at the request of R. K. Guy, Sep 03 2010

The member k=11 of a family of sequences starting 0,1,1,k with recurrence a(n) = a(n-1)+k*a(n-2)+a(n-3)-a(n-4).

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Hugh Williams, R. K. Guy, Some fourth-order linear divisibility sequences, Intl. J. Number Theory vol. 7 (5) (2011) 1255-1277
Index entries for linear recurrences with constant coefficients, signature (1,11,1,-1).

Cf. A116201 (k=1), A105309 (k=2), A152090 (k=3), A007598 (k=4), A005178 (k=5), A003757 (k=6).

I:=[0, 1, 1, 11]; [n le 4 select I[n] else Self(n-1) + 11*Self(n-2) + Self(n-3) - Self(n-4): n in [1..30]]; // Vincenzo Librandi, Dec 19 2012

CoefficientList[Series[-x*(x - 1)*(1 + x)/(1 - x - 11*x^2 - x^3 + x^4), {x, 0, 40}], x] (* Vincenzo Librandi, Dec 19 2012 *)

a(n)= +a(n-1) +11*a(n-2) +a(n-3) -a(n-4).

cp's OEIS Frontend

User: R. K. Guy

A259472 Coefficients in an asymptotic expansion of A003319(n)/n! in falling factorials.

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A242497 Sides of (Heronian) triangles where sides are consecutive integers and area is an integer.

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A214525 a(n) = 7*a(n-1) - 23*a(n-2) + 49*a(n-3) - 49*a(n-4) with a(0)=0, a(1)=1, a(2)=7, a(3)=19.

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A212804 Expansion of (1 - x)/(1 - x - x^2).

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A171099 a(n) = number of solutions (x,y) (with 0 <= x <= y) to x*(x+1)/2 + y*(y+1)/2 = n!.

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A171064 G.f.: -x*(x-1)*(1+x)/(1-x-7*x^2-x^3+x^4).

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A171065 G.f. -x*(x-1)*(1+x)/(1-x-8*x^2-x^3+x^4).

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A214525 a(n) = 7a(n-1) - 23a(n-2) + 49a(n-3) - 49a(n-4) with a(0)=0, a(1)=1, a(2)=7, a(3)=19.

A171099 a(n) = number of solutions (x,y) (with 0 <= x <= y) to x(x+1)/2 + y(y+1)/2 = n!.

A171064 G.f.: -x(x-1)(1+x)/(1-x-7*x^2-x^3+x^4).

A171065 G.f. -x(x-1)(1+x)/(1-x-8*x^2-x^3+x^4).

A171066 G.f. -x(x-1)(1+x)/(1-x-9*x^2-x^3+x^4).

A171067 G.f. -x(x-1)(1+x)/((x^2+3x+1)(x^2-4*x+1)).

A171068 G.f. -x(x-1)(1+x)/(1-x-11*x^2-x^3+x^4).