A071675 -id:A071675 - cp's OEIS frontend

A103778 Inverse of trinomial triangle A071675.

1, 0, 1, 0, -1, 1, 0, 1, -2, 1, 0, 0, 3, -3, 1, 0, -4, -2, 6, -4, 1, 0, 14, -7, -7, 10, -5, 1, 0, -30, 36, -6, -16, 15, -6, 1, 0, 33, -96, 63, 3, -30, 21, -7, 1, 0, 55, 154, -209, 88, 25, -50, 28, -8, 1, 0, -429, 0, 429, -374, 99, 66, -77, 36, -9, 1, 0, 1365, -1014

Offset: 0

Paul Barry, Feb 15 2005

Inverse of Riordan array (1, x(1+x+x^2)) (A071675).

			Rows start
{1},
{0,1},
{0,-1,1},
{0,1,-2,1},
{0,0,3,-3,1},
...

Cf. A071675.

Riordan array (1, y) where y satisfies y + y^2 + y^3 = x; y = -2^(2/3)*((3*sqrt(3)*sqrt(27*x^2 + 14*x + 3) - 27*x - 7)^(1/3) - (3*sqrt(3)*sqrt(27*x^2 + 14*x + 3) + 27*x + 7)^(1/3) + 2^(1/3))/6.

A103780 Row sums of square of trinomial triangle A071675.

Original entry on oeis.org

1, 1, 3, 9, 25, 69, 189, 519, 1428, 3930, 10812, 29742, 81816, 225070, 619156, 1703262, 4685565, 12889687, 35458707, 97544655, 268339161, 738183999, 2030697309, 5586319365, 15367609920, 42275319276, 116296719448

Offset: 0

Paul Barry, Feb 15 2005

G. C. Greubel, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (1,2,4,6,8,8,6,3,1).

CoefficientList[Series[1/(1 - x - 2*x^2 - 4*x^3 - 6*x^4 - 8*x^5 - 8*x^6 - 6*x^7 - 3*x^8 - x^9), {x,0,50}], x] (* G. C. Greubel, Mar 03 2017 *)
LinearRecurrence[{1,2,4,6,8,8,6,3,1},{1,1,3,9,25,69,189,519,1428},40] (* Harvey P. Dale, Jun 14 2020 *)

a(n):=sum(sum((sum(binomial(j,n-3*k+2*j)*(-1)^(j-k)*binomial(k,j),j,0,k)) *sum(binomial(j,-3*m+k+2*j)*binomial(m,j),j,0,m),k,m,n),m,0,n); /* Vladimir Kruchinin, Dec 01 2011 */

x='x+O('x^50); Vec(1/(1 -x -2*x^2 -4*x^3 -6*x^4 -8*x^5 -8*x^6 -6*x^7 -3*x^8 -x^9)) \\ G. C. Greubel, Mar 03 2017

G.f.: 1/(1-x-2*x^2-4*x^3-6*x^4-8*x^5-8*x^6-6*x^7-3*x^8-x^9).

a(n) = a(n-1) +2a(n-2) +4a(n-3) +6a(n-4) +8a(n-5) +8a(n-6) +6a(n-7) +3a(n-8) +a(n-9).

A098975 Nonzero elements of table A071675; also counts selected ordered multisets of the values {1,2,3}.

Original entry on oeis.org

1, 1, 1, 1, 1, 2, 1, 3, 3, 1, 2, 6, 4, 1, 1, 7, 10, 5, 1, 6, 16, 15, 6, 1, 3, 19, 30, 21, 7, 1, 1, 16, 45, 50, 28, 8, 1, 10, 51, 90, 77, 36, 9, 1, 4, 45, 126, 161, 112, 45, 10, 1, 1, 30, 141, 266, 266

Offset: 0

Alford Arnold, Oct 23 2004

			Shift A027907 as indicated and construct a(n) by column.
The table begins
  1
  . 1 1 1
  . . 1 2 3 2 1
  . . . 1 3 6 7 6 3 1
  . . . . 1 4 10 16 19 16 10 4 1
  . . . . . 1 5 15 30 45 51 45 30 15 5 1
  . . . . . . 1
therefore a(19) = 6 and appears in column seven and row three.
a(19) counts the ordered multisets {133,313,331,223,232,322}.

Cf. A000073, A027907, A071675, A098925.

A013979 Expansion of 1/(1 - x^2 - x^3 - x^4) = 1/((1 + x)*(1 - x - x^3)).

Original entry on oeis.org

1, 0, 1, 1, 2, 2, 4, 5, 8, 11, 17, 24, 36, 52, 77, 112, 165, 241, 354, 518, 760, 1113, 1632, 2391, 3505, 5136, 7528, 11032, 16169, 23696, 34729, 50897, 74594, 109322, 160220, 234813, 344136, 504355, 739169, 1083304, 1587660, 2326828, 3410133, 4997792, 7324621

Offset: 0

N. J. A. Sloane

For n>0, number of compositions (ordered partitions) of n into 2's, 3's and 4's. - Len Smiley, May 08 2001
Diagonal sums of trinomial triangle A071675 (Riordan array (1, x*(1+x+x^2))). - Paul Barry, Feb 15 2005
For n>1, a(n) is number of compositions of n-2 into parts 1 and 2 with no 3 consecutive 1's. For example: a(7) = 5 because we have: 2+2+1, 2+1+2, 1+2+2, 1+2+1+1, 1+1+2+1. - Geoffrey Critzer, Mar 15 2014
In the same way [per 2nd comment for A006498, by Sreyas Srinivasan] that the sum of any two alternating terms (terms separated by one term) of A006498 produces a term from A000045 (the Fibonacci sequence), so it could therefore be thought of as a "metaFibonacci," the sum of any two (nonalternating) terms of this sequence produces a term from A000930 (Narayana’s cows), so this sequence could analogously be called "meta-Narayana’s cows" (e.g. 4+5=9, 5+8=13, 8+11=19, 11+17=28). - Michael Cohen and Yasuyuki Kachi, Jun 13 2024

			G.f. = 1 + x^2 + x^3 + 2*x^4 + 2*x^5 + 4*x^6 + 5*x^7 + 8*x^8 + 11*x^9 + ...

Reinhard Zumkeller, Table of n, a(n) for n = 0..1000
Michael Cohen and Yasuyuki Kachi, Recurrence Relations Rhythm. In: Noll, T., Montiel, M., Gómez, F., Hamido, O.C., Besada, J.L., Martins, J.O. (eds) Mathematics and Computation in Music. MCM 2024. Lecture Notes in Computer Science, vol. 14639. Springer, Cham.
C. K. Fan, A Hecke algebra quotient and some combinatorial applications, J. Algebraic Combin. 5 (1996), no. 3, 175-189.
C. K. Fan, Structure of a Hecke algebra quotient, J. Amer. Math. Soc. 10 (1997), no. 1, 139-167. [Page 156, f^0_n.]
R. Mullen, On Determining Paint by Numbers Puzzles with Nonunique Solutions, JIS 12 (2009) 09.6.5.
Index entries for linear recurrences with constant coefficients, signature (0,1,1,1).

Cf. A060945 (Ordered partitions into 1's, 2's and 4's).
First differences of A023435.
Cf. A000045, A000930, A001634, A006498, A071675, A077889, A078012, A097333, A107458.

a013979 n = a013979_list !! n
a013979_list = 1 : 0 : 1 : 1 : zipWith (+) a013979_list
   (zipWith (+) (tail a013979_list) (drop 2 a013979_list))
-- Reinhard Zumkeller, Mar 23 2012

R:=PowerSeriesRing(Integers(), 50); Coefficients(R!( 1/((1+x)*(1-x-x^3)) )); // G. C. Greubel, Jul 17 2023

a[n_]:= If[n<0, SeriesCoefficient[x^4/(1 +x +x^2 -x^4), {x, 0, -n}], SeriesCoefficient[1/(1 -x^2 -x^3 -x^4), {x,0,n}]]; (* Michael Somos, Jun 20 2015 *)
LinearRecurrence[{0,1,1,1}, {1,0,1,1}, 50] (* G. C. Greubel, Jul 17 2023 *)

@CachedFunction
def b(n): return 1 if (n<3) else b(n-1) + b(n-3) # b = A000930
def A013979(n): return ((-1)^n +2*b(n) -b(n-1) +b(n-2) -int(n==1))/3
[A013979(n) for n in (0..50)] # G. C. Greubel, Jul 17 2023

a(n) = Sum_{k=0..floor(n/2)} Sum_{i=0..floor(n/2)} C(k, 2i+3k-n)*C(2i+3k-n, i). - Paul Barry, Feb 15 2005
a(n) = a(n-4) + a(n-3) + a(n-2). - Jon E. Schoenfield, Aug 07 2006
a(n) + a(n+1) = A000930(n+1). - R. J. Mathar, Mar 14 2011
a(n) = (1/3)*(A000930(n) + A097333(n-2) + (-1)^n), n>1. - Ralf Stephan, Aug 15 2013
a(n) = (-1)^n * A077889(-4-n) = A107458(n+4) for all n in Z. - Michael Somos, Jun 20 2015
a(n) = Sum_{i=0..floor(n/2)} A078012(n-2*i). - Paul Curtz, Aug 18 2021
a(n) = (1/3)*((-1)^n + 2*b(n) - b(n-1) + b(n-2) - [n=1]), where b(n) = A000930(n). - G. C. Greubel, Jul 17 2023

A213887 Triangle of coefficients of representations of columns of A213743 in binomial basis.

Original entry on oeis.org

1, 0, 1, 0, 1, 1, 0, 1, 2, 1, 0, 1, 3, 3, 1, 0, 0, 4, 6, 4, 1, 0, 0, 3, 10, 10, 5, 1, 0, 0, 2, 12, 20, 15, 6, 1, 0, 0, 1, 12, 31, 35, 21, 7, 1, 0, 0, 0, 10, 40, 65, 56, 28, 8, 1, 0, 0, 0, 6, 44, 101, 120, 84, 36, 9, 1, 0

Offset: 0

Vladimir Shevelev and Peter J. C. Moses, Jun 23 2012

This triangle is the third array in the sequence of arrays A026729, A071675 considered as triangles.
Let {a_(k,i)}, k>=1, i=0,...,k, be the k-th row of the triangle. Then s_k(n)=sum{i=0,...,k}a_(k,i)* binomial(n,k) is the n-th element of the k-th column of A213743. For example, s_1(n)=binomial(n,1)=n is the first column of A213743 for n>1, s_2(n)=binomial(n,1)+binomial(n,2)is the second column of A213743 for n>1, etc. In particular (see comment in A213743), in cases k=6,7,8,9  s_k(n) is A064056(n+2), A064057(n+2), A064058(n+2), A000575(n+3) respectively.
Riordan array (1,x+x^2+x^3+x^4). A186332 with additional 0 column. - Ralf Stephan, Dec 31 2013

			As a triangle, this begins
n/k.|..0....1....2....3....4....5....6....7....8....9
=====================================================
.0..|..1
.1..|..0....1
.2..|..0....1....1
.3..|..0....1....2....1
.4..|..0....1....3....3....1
.5..|..0....0....4....6....4....1
.6..|..0....0....3...10...10....5....1
.7..|..0....0....2...12...20...15....6....1
.8..|..0....0....1...12...31...35...21....7....1
.9..|..0....0....0...10...40...65...56...28....8....1

Cf. A026729, A071675, A030528 (parts <=2), A078803 (parts <=3), A213888 (parts <=5), A061676 and A213889 (parts <=6).

pts := 4; # A213887
g := 1/(1-t*z*add(z^i,i=0..pts-1)) ;
for n from 0 to 13 do
    for k from 0 to n do
        coeftayl(g,z=0,n) ;
        coeftayl(%,t=0,k) ;
        printf("%d ",%) ;
    end do:
    printf("\n") ;
end do: # R. J. Mathar, May 28 2025

A213888 Triangle of coefficients of representations of columns of A213744 in binomial basis.

Original entry on oeis.org

1, 0, 1, 0, 1, 1, 0, 1, 2, 1, 0, 1, 3, 3, 1, 0, 1, 4, 6, 4, 1, 0, 0, 5, 10, 10, 5, 1, 0, 0, 4, 15, 20, 15, 6, 1, 0, 0, 3, 18, 35, 35, 21, 7, 1, 0, 0, 2, 19, 52, 70, 56, 28, 8, 1, 0, 0, 1, 18, 68, 121, 126, 84, 36, 9, 1, 0

Offset: 0

Vladimir Shevelev and Peter J. C. Moses, Jun 23 2012

This triangle is the fourth array in the sequence of arrays A026729, A071675, A213887,..., such that the first two arrays are considered as triangles.

Let {a_(k,i)}, k>=1, i=0,...,k, be the k-th row of the triangle. Then s_k(n)=sum{i=0,...,k}a_(k,i)* binomial(n,k) is the n-th element of the k-th column of A213744. For example, s_1(n)=binomial(n,1)=n is the first column of A213744 for n>1, s_2(n)=binomial(n,1)+binomial(n,2)is the second column of A213744 for n>1, etc. In particular (see comment inA213744), in cases k=7,8,9 s_k(n) is A063262(n+2), A063263(n+2), A063264(n+2) respectively.

			As a triangle, this begins
n/k.|..0....1....2....3....4....5....6....7....8....9
=====================================================
.0..|..1
.1..|..0....1
.2..|..0....1....1
.3..|..0....1....2....1
.4..|..0....1....3....3....1
.5..|..0....1....4....6....4....1
.6..|..0....0....5...10...10....5....1
.7..|..0....0....4...15...20...15....6....1
.8..|..0....0....3...18...35...35...21....7....1
.9..|..0....0....2...19...52...70...56...28....8....1

Cf. A026729, A071675, A213887, A030528 (parts <=2), A078803 (parts <=3), A213887 (parts <=4).

pts := 5; # A213888
g := 1/(1-t*z*add(z^i,i=0..pts-1)) ;
for n from 0 to 13 do
    for k from 0 to n do
        coeftayl(g,z=0,n) ;
        coeftayl(%,t=0,k) ;
        printf("%d ",%) ;
    end do:
    printf("\n") ;
end do: # R. J. Mathar, May 28 2025

A005718 Quadrinomial coefficients: C(2+n,n) + C(3+n,n) + C(4+n,n).

Original entry on oeis.org

3, 12, 31, 65, 120, 203, 322, 486, 705, 990, 1353, 1807, 2366, 3045, 3860, 4828, 5967, 7296, 8835, 10605, 12628, 14927, 17526, 20450, 23725, 27378, 31437, 35931, 40890, 46345, 52328, 58872, 66011, 73780, 82215, 91353, 101232, 111891, 123370, 135710, 148953, 163142, 178321, 194535

Offset: 0

N. J. A. Sloane

If Y is an (n-3)-subset of an n-set X then, for n>=5, a(n-5) is the number of 4-subsets of X having at least two elements in common with Y. - Milan Janjic, Dec 16 2007
This equation represents the number of numbers with <=n digits such that the sum of the digits is between 1 and 4 inclusive and no digit is larger than 3. - David Consiglio, Jr., Oct 27 2008
Row 2 of the convolution array A213548. - Clark Kimberling, Jun 20 2012

L. Comtet, Advanced Combinatorics, Reidel, 1974, p. 78.
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
Niall Byrnes, Gary R. W. Greaves, and Matthew R. Foreman, Bootstrapping cascaded random matrix models: correlations in permutations of matrix products, arXiv:2405.02541 [math-ph], 2024. See p. 7.
R. K. Guy, Letter to N. J. A. Sloane, 1987.
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.
Index entries for linear recurrences with constant coefficients, signature (5,-10,10,-5,1).

Cf. A008287, A062745, A063421, A071675, A213548.

[(((n+14)*n+71)*n+130)*n/24+3: n in [0..45]]; // Vincenzo Librandi, Jun 15 2011

A005718:=-(3-3*z+z**2)/(z-1)**5; # conjectured by Simon Plouffe in his 1992 dissertation

Table[Plus@@Table[Binomial[i + n, n], {i, 2, 4}], {n, 0, 43}] (* From Alonso del Arte, Jun 14 2011 *)

a(n)=(((n+14)*n+71)*n+130)*n/24+3 \\ Charles R Greathouse IV, Jun 14 2011

a(n) = binomial(n, 2)*(n^2+7*n+18)/12, n >= 2.
G.f.: (3-3*x+x^2)/(1-x)^5. (numerator polynomial is N4(4, x) from A063421).
a(n) = A008287(n, 4), n >= 2 (fifth column of quadrinomial coefficients).
a(n) = A062745(n, 4), n >= 2 (fifth column).
a(n) = 3*C(n+2,2) + 3*C(n+2,3) + C(n+2,4) (see comment in A071675). - Vladimir Shevelev and Peter J. C. Moses, Jun 22 2012
E.g.f.: exp(x)*(72 + 216*x + 120*x^2 + 20*x^3 + x^4)/24. - Stefano Spezia, May 09 2024

Better description from Zerinvary Lajos, Dec 02 2005

A077835 Expansion of 1/(1 - 2x - 2x^2 - 2*x^3).

Original entry on oeis.org

1, 2, 6, 18, 52, 152, 444, 1296, 3784, 11048, 32256, 94176, 274960, 802784, 2343840, 6843168, 19979584, 58333184, 170311872, 497249280, 1451788672, 4238699648, 12375475200, 36131927040, 105492203776, 307999212032, 899246685696, 2625476203008, 7665444201472

Offset: 0

N. J. A. Sloane, Nov 17 2002

a(n) is the number of ways two opposing basketball teams could score a combined total of n points (counting one point free throws, two point field goals, and three point field goals) considering the order of the scoring as important. - Geoffrey Critzer, Feb 07 2009
Number of permutations of length a(n+1) avoiding the partially ordered pattern (POP) {1>3, 4>2} of length 4. That is, number of length n permutations having no subsequences of length 4 in which the first element is larger than the third element, and the fourth element is larger than the second element. - Sergey Kitaev, Dec 08 2020
a(n) is the number of compositions of n into parts 1, 3, and 3, each part of two kinds. - Joerg Arndt, Jul 30 2023

Michael De Vlieger, Table of n, a(n) for n = 0..2149
Martin Burtscher, Igor Szczyrba, and Rafał Szczyrba, Analytic Representations of the n-anacci Constants and Generalizations Thereof, Journal of Integer Sequences, Vol. 18 (2015), Article 15.4.5.
Alice L. L. Gao and Sergey Kitaev, On partially ordered patterns of length 4 and 5 in permutations, arXiv:1903.08946 [math.CO], 2019.
Alice L. L. Gao and Sergey Kitaev, On partially ordered patterns of length 4 and 5 in permutations, The Electronic Journal of Combinatorics 26(3) (2019), P3.26.
Yassine Otmani, The 2-Pascal Triangle and a Related Riordan Array, J. Int. Seq. (2025) Vol. 28, Issue 3, Art. No. 25.3.5. See p. 19.
Index entries for linear recurrences with constant coefficients, signature (2,2,2).

Cf. A071675.

LinearRecurrence[{2, 2, 2}, {1, 2, 6}, 100] (* Vladimir Joseph Stephan Orlovsky, Jul 03 2011 *)
m={{2/3,1/3,0,0},{2/3,0,1/3,0},{2/3,0,0,1/3},{0,0,0,0}};
initialState={{1,0,0,0}};
Table[(initialState.MatrixPower[m,n])[[1,4]]*3^n,{n,3,31}] (* Robert P. P. McKone, Jul 29 2023 *)

Vec(1/(1-2*x-2*x^2-2*x^3)+O(x^99)) \\ Charles R Greathouse IV, Sep 24 2012

a(n) = Sum_{k=0..n} T(n-k, k)*2^(n-k), T(n, k) = trinomial coefficients (A027907). - Paul Barry, Feb 15 2005
a(n) = Sum_{k=0..n} 2^k * Sum_{i=0..floor((n-k)/2)} C(n-k-i, i)*C(k, n-k-i). - Paul Barry, Apr 26 2005
a(n) = 2*a(n-1) + 2*a(n-2) + 2*a(n-3). - Geoffrey Critzer, Feb 07 2009

A078803 Triangular array T given by T(n,k) = number of compositions of n into k parts, each in the set {1,2,3}.

Original entry on oeis.org

1, 1, 1, 1, 2, 1, 0, 3, 3, 1, 0, 2, 6, 4, 1, 0, 1, 7, 10, 5, 1, 0, 0, 6, 16, 15, 6, 1, 0, 0, 3, 19, 30, 21, 7, 1, 0, 0, 1, 16, 45, 50, 28, 8, 1, 0, 0, 0, 10, 51, 90, 77, 36, 9, 1, 0, 0, 0, 4, 45, 126, 161, 112, 45, 10, 1, 0, 0, 0, 1, 30, 141, 266, 266, 156, 55, 11, 1, 0, 0, 0, 0, 15, 126

Offset: 1

Clark Kimberling, Dec 06 2002

Number of lattice paths from (0,0) to (n,k) using steps (1,1), (2,1), (3,1). - Joerg Arndt, Jul 05 2011
Reversing the rows produces A078802. Row sums: A000073.
Number of tribonacci binary words of length n-1 having k-1 1's. A tribonacci binary word is a binary word having no three consecutive 0's. Example: T(6,3)=7 because we have 00101,00110,01001,01010,01100,10010 and 10100. - Emeric Deutsch, Jun 16 2007
This is the Riordan array (1,x+x^2+x^3)(A071675) without its column k=0. - Vladimir Kruchinin, Feb 10 2011

			T(5,2) = 2 counts the compositions 2+3 and 3+2.
Triangle begins
  1;
  1, 1;
  1, 2, 1;
  0, 3, 3, 1;
  0, 2, 6, 4, 1;
  0, 1, 7, 10, 5, 1;
  0, 0, 6, 16, 15, 6, 1;
  0, 0, 3, 19, 30, 21, 7, 1;
  0, 0, 1, 16, 45, 50, 28, 8, 1;
  0, 0, 0, 10, 51, 90, 77, 36, 9, 1;
  0, 0, 0, 4, 45, 126, 161, 112, 45, 10, 1;
  0, 0, 0, 1, 30, 141, 266, 266, 156, 55, 11, 1;

Clark Kimberling, Binary words with restricted repetitions and associated compositions of integers, in Applications of Fibonacci Numbers, vol.10, Proceedings of the Eleventh International Conference on Fibonacci Numbers and Their Applications, William Webb, editor, Congressus Numerantium, Winnipeg, Manitoba 194 (2009) 141-151.

Alois P. Heinz, Rows n = 1..141, flattened
Jean Luc Baril, Rigoberto Flórez, and José L. Ramirez, Generalized Narayana arrays, restricted Dyck paths, and related bijections, Univ. Bourgogne (France, 2025). See p. 18.
Vladimir Kruchinin, Composition of ordinary generating functions, arXiv:1009.2565 [math.CO], 2010.

Cf. A027907, A078802, A030528 (parts <=2), A213887 (parts <=4), A213888 (parts <=5), A061676 and A213889 (parts <=6).

A078803 := proc(n,k) add( binomial(j,n-3*k+2*j)*binomial(k,j),j=0..k) ; end proc:
# R. J. Mathar, Feb 22 2011

nn=8;CoefficientList[Series[1/(1-y(x+x^2+x^3)),{x,0,nn}],{x,y}]//Grid (* Geoffrey Critzer, Jan 08 2013 *)

T(n, k) = t(n-1, n-k), for 1<=k<=n, for n>=1, where the array t is given by A078802.
G.f.: 1/(1-t*z*(1+z+z^2))-1. - Emeric Deutsch, Mar 10 2004
T(n,k) = Sum_{j=0..k} C(j,n-3*k+2*j)*C(k,j). - Vladimir Kruchinin, Feb 10 2011

More terms from Emeric Deutsch, Jun 16 2007

A001919 Eighth column of quadrinomial coefficients.

Original entry on oeis.org

6, 40, 155, 456, 1128, 2472, 4950, 9240, 16302, 27456, 44473, 69680, 106080, 157488, 228684, 325584, 455430, 627000, 850839, 1139512, 1507880, 1973400, 2556450, 3280680, 4173390, 5265936, 6594165, 8198880, 10126336, 12428768, 15164952, 18400800, 22209990

Offset: 3

N. J. A. Sloane

L. Comtet, Advanced Combinatorics, Reidel, 1974, p. 78.
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).

T. D. Noe, Table of n, a(n) for n = 3..1000
L. Carlitz et al., Permutations and sequences with repetitions by number of increases, J. Combin. Theory, 1 (1966), 350-374.
R. K. Guy, Letter to N. J. A. Sloane, 1987
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
Index entries for linear recurrences with constant coefficients, signature (8, -28, 56, -70, 56, -28, 8, -1).

seq(n*(n^2-1)*(n^2-4)*(n^2+21*n+180)/5040,n=3..34); # Emeric Deutsch, Jan 27 2005
A001919:=(3*z**2-8*z+6)/(z-1)**8; # conjectured by Simon Plouffe in his 1992 dissertation

Table[n*(n^2 - 1)*(n^2 - 4)*(n^2 + 21*n + 180)/5040, {n, 3, 50}] (* T. D. Noe, Aug 17 2012 *)
LinearRecurrence[{8,-28,56,-70,56,-28,8,-1},{6,40,155,456,1128,2472,4950,9240},40] (* Harvey P. Dale, Mar 27 2013 *)

a(n) = A008287(n, 7) = binomial(n+2, 5)*(n^2+21*n+180 )/42, n >= 3.
G.f.: (x^3)*(6-8*x+3*x^2 )/(1-x)^8. Numerator polynomial is N4(7, x) from array A063421.
a(n) = n(n^2-1)(n^2-4)(n^2+21n+180)/5040. - Emeric Deutsch, Jan 27 2005
a(n) = 6*C(n,3) + 16*C(n,4) + 15*C(n,5) + 6*C(n,6) + C(n,7) (see comment in A071675). - Vladimir Shevelev and Peter J. C. Moses, Jun 22 2012
a(3)=6, a(4)=40, a(5)=155, a(6)=456, a(7)=1128, a(8)=2472, a(9)=4950, a(10)=9240, 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). - Harvey P. Dale, Mar 27 2013

More terms from Emeric Deutsch, Jan 27 2005

cp's OEIS Frontend

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A213887 Triangle of coefficients of representations of columns of A213743 in binomial basis.

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A077835 Expansion of 1/(1 - 2x - 2x^2 - 2*x^3).