A103142 -id:A103142 - cp's OEIS frontend

A103141 Riordan array (1/(1-x), x*(1 + x + x^2 + x^3)/(1-x)).

1, 1, 1, 1, 3, 1, 1, 6, 5, 1, 1, 10, 15, 7, 1, 1, 14, 35, 28, 9, 1, 1, 18, 68, 84, 45, 11, 1, 1, 22, 116, 207, 165, 66, 13, 1, 1, 26, 180, 441, 491, 286, 91, 15, 1, 1, 30, 260, 840, 1251, 996, 455, 120, 17, 1, 1, 34, 356, 1464, 2823, 2948, 1814, 680, 153, 19, 1

Offset: 0

Paul Barry, Jan 24 2005

Generalized Pascal matrix: row sums are generalized Pell numbers A103142 and diagonal sums are the Pentanacci numbers A001591(n+4). One of a family of generalized Pascal triangles given by the Riordan arrays (1/(1-x), x*Sum_{j=0..k} x^k/(1-x)). This array has the 'k+2-nacci' numbers as diagonal sums and generalized Pell numbers b(n) = 2b(n-1) + Sum_{j=1..k} b(n-1-j) as row sums. The first two arrays of the family are Pascal's triangle and the Delannoy number triangle.

			Triangle begins
  1;
  1,  1;
  1,  3,   1;
  1,  6,   5,    1;
  1, 10,  15,    7,    1;
  1, 14,  35,   28,    9,    1;
  1, 18,  68,   84,   45,   11,    1;
  1, 22, 116,  207,  165,   66,   13,   1;
  1, 26, 180,  441,  491,  286,   91,  15,   1;
  1, 30, 260,  840, 1251,  996,  455, 120,  17,  1;
  1, 34, 356, 1464, 2823, 2948, 1814, 680, 153, 19, 1; ...

G. C. Greubel, Rows n = 0..100 of triangle, flattened
Kuhapatanakul, Kantaphon; Anantakitpaisal, Pornpawee The k-nacci triangle and applications. Cogent Math. 4, Article ID 1333293, 13 p. (2017).

Cf. A102036.

T[?Positive, 0] = 1; T[n, n_] = 1; T[n_, k_] /; 0, ] = 0; Table[T[n, k], {n, 0, 11}, {k, 0, n}] // Flatten (* Jean-François Alcover, Apr 24 2017 *)

T(n,k)=polcoef(polcoef(1/(1-x-x*y*(1+x+x^2+x^3)) + O(x*x^n), n), k) \\ Andrew Howroyd, Dec 12 2018

def A103141Triangle(dim):
    def B(n): return n if n < 5 else 4
    M = matrix(ZZ, dim, dim)
    for k in (0..dim-1): M[k, 0] = 1
    for k in (1..dim-1):
        for m in (k..dim-1):
            M[m, k] = sum(M[j, k-1]*B(m-j) for j in (k-1..m-1))
    return M
A103141Triangle(11) # Peter Luschny, Dec 22 2018

Triangle, read by rows, where the terms are generated by the rule: T(n, k) = T(n-1, k) + T(n-1, k-1) + T(n-2, k-1) + T(n-3, k-1) + T(n-4, k-1), with T(0, 0)=1.
G.f.: 1/(1-x-x*y*(1+x+x^2+x^3)). - Vladimir Kruchinin, Apr 21 2015
From Werner Schulte, Dec 07 2018, Dec 12 2018, Dec 13 2018: (Start)
G.f. of column k: Sum_{n>=0} T(n+k,k) * x^n = (1+x+x^2+x^3)^k / (1-x)^(k+1) = (1-x^4)^k / (1-x)^(2*k+1).
Let k >= 0 be some fixed integer and a_k(n) be multiplicative with a_k(p^e) = T(e+k,k) for prime p and e >= 0. Then we have the Dirichlet g.f.: Sum{n>0} a_k(n) / n^s = (zeta(s))^(2*k+1) / (zeta(4*s))^k.
T(n,k) = Sum_{i=0..n-k} binomial(n-i,k) * (Sum_{j=0..i} binomial(k,j) * binomial(3*k-2*j,i-j) * (-2)^j) for 0 <= k <= n (conjectured).
T(n,k) = Sum_{i=0..n-k} binomial(n-i,k) * (Sum_{j=0..floor(i/4)} (-1)^j * binomial(k,j) * binomial(k-1+i-4*j,i-4*j)) for 0 <= k <= n.
T(n,k) = Sum_{i=0..n-k} binomial(n-i,k) * (Sum_{j=0..floor(i/2)} binomial(k,j) * binomial(k,i-2*j)) for 0 <= k <= n. (End)

A141448 Generalized Pell numbers P(n,5,5).

Original entry on oeis.org

0, 1, 2, 5, 13, 34, 89, 232, 605, 1578, 4116, 10736, 28003, 73041, 190515, 496926, 1296147, 3380779, 8818187, 23000741, 59993521, 156482896, 408159020, 1064613385, 2776862948, 7242974718, 18892067685, 49276745441, 128530009618

Offset: 0

R. J. Mathar, Aug 07 2008

P(n,2,2) and P(n,2,1) are in A000129.
P(n,3,2) is A116413. P(n,3,1) and P(n,3,3) are A077939.
P(n,4,1) and P(n,4,4) are A103142.

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Brian Hopkins and Stéphane Ouvry, Combinatorics of Multicompositions, arXiv:2008.04937 [math.CO], 2020.
E. Kilic and D. Tasci, The generalized Binet formula, representation and sums of the generalizedorder-k Pell numbers, Taiwanese J of Math vol 10 no 6 (2006), 1661-1670.
Index entries for linear recurrences with constant coefficients, signature (2,1,1,1,1).

I:=[0,1,2,5,13]; [n le 5 select I[n] else 2*Self(n-1)+Self(n-2)+Self(n-3)+Self(n-4)+Self(n-5): n in [1..30]]; // Vincenzo Librandi, Dec 13 2012

P := proc(n,k,i) option remember ; if n = 1-i then 1; elif n <= 0 then 0; else 2*P(n-1,k,i)+add(P(n-j,k,i),j=2..k) ; fi ; end: for n from 0 to 40 do printf("%d,",P(n,5,5)) ; od:

CoefficientList[Series[x/(1 - 2*x - x^2 - x^3 - x^4 - x^5), {x, 0, 40}], x] (* Vincenzo Librandi, Dec 13 2012 *)
LinearRecurrence[{2,1,1,1,1},{0,1,2,5,13},40] (* Harvey P. Dale, Jan 08 2016 *)

a(n):=b(n+1);
b(n):=sum(sum(binomial(k,r)*2^(k-r)*sum((sum(binomial(j,-r+n-m-k-j)*binomial(m,j),j,0,m))*binomial(r,m),m,0,r),r,0,k),k,1,n); /* Vladimir Kruchinin, May 05 2011 */

From R. J. Mathar, Nov 28 2008: (Start)
a(n) = 2*a(n-1) + a(n-2) + a(n-3) + a(n-4) + a(n-5).
G.f.: x/(1-2*x-x^2-x^3-x^4-x^5). (End)
a(n+1) = Sum_(k=1..n, Sum_(r=0..k, binomial(k,r)*2^(k-r)*Sum_(m=0..r,(Sum_(j=0..m, binomial(j,-r+n-m-k-j)*binomial(m,j)))*binomial(r,m)))), a(0)=0, a(1)=1. [Vladimir Kruchinin, May 05 2011]

A190139 a(n) = 2*a(n-1) + a(n-2) + a(n-3) + a(n-4), a(-2)=0, a(-1)=0, a(0)=1, a(1)=1.

Original entry on oeis.org

1, 1, 3, 8, 21, 54, 140, 363, 941, 2439, 6322, 16387, 42476, 110100, 285385, 739733, 1917427, 4970072, 12882689, 33392610, 86555408, 224356187, 581543081, 1507390367, 3907235410, 10127760455, 26251689768, 68045765768, 176378217169, 457181650329, 1185038973363

Offset: 0

Vladimir Kruchinin, May 05 2011

nonn

Number of compositions of n where there is one sort of part 1, two sorts of part 2, three sorts of part 3, and four sorts of every other part. - Joerg Arndt, Mar 15 2014

G. C. Greubel, Table of n, a(n) for n = 0..1000 (terms 0..200 from Vincenzo Librandi)
Index entries for linear recurrences with constant coefficients, signature (2,1,1,1)

I:=[1,1,3,8]; [n le 4 select I[n] else 2*Self(n-1)+Self(n-2)+Self(n-3)+Self(n-4): n in [1..40]]; // Vincenzo Librandi, Sep 20 2011

RecurrenceTable[{a[n] == 2 a[n - 1] + a[n - 2] + a[n - 3] + a[n - 4], a[-2] == a[-1] == 0, a[0] == a[1] == 1}, a, {n, 0, 30}] (* Michael De Vlieger, Oct 28 2015 *)
LinearRecurrence[{2, 1, 1, 1}, {1, 1, 3, 8}, 31] (* Michael De Vlieger, Oct 28 2015 *)
nxt[{a_,b_,c_,d_}]:={b,c,d,2d+c+b+a}; NestList[nxt,{0,0,1,1},50][[All,1]] (* Harvey P. Dale, Mar 04 2022 *)

a(n):=sum(sum((sum(binomial(k,j)*sum(binomial(j,i-j)*binomial(k-j,t-3*(k-j)-i),i,j,t-k+j),j,0,k))*binomial(-t+n+k-1,k-1),t,k,n),k,1,n);

x='x+O('x^30); Vec((1-x)/(1-2*x-x^2-x^3-x^4)) \\ G. C. Greubel, Dec 29 2017

G.f.: (1-x)/(1-2*x-x^2-x^3-x^4).
a(n) = Sum_{k=1..n} (Sum_{t=k..n} (Sum_{j=0..k} C(k,j) * Sum_{i=j..t-k+j} C(j,i-j)*C(k-j,t-3*(k-j)-i)*C(-t+n+k-1,k-1))), n>0, a(0)=1.
a(n) = A103142(n) - A103142(n-1). - R. J. Mathar, Sep 17 2011

Deleted certain dangerous or potentially dangerous links. - N. J. A. Sloane, Jan 30 2021

A202193 Triangle read by rows: T(n,m) = coefficient of x^n in expansion of (x/(1 - x - x^2 - x^3 - x^4))^m.

Original entry on oeis.org

1, 1, 1, 2, 2, 1, 4, 5, 3, 1, 8, 12, 9, 4, 1, 15, 28, 25, 14, 5, 1, 29, 62, 66, 44, 20, 6, 1, 56, 136, 165, 129, 70, 27, 7, 1, 108, 294, 401, 356, 225, 104, 35, 8, 1, 208, 628, 951, 944, 676, 363, 147, 44, 9, 1, 401, 1328, 2211, 2424, 1935, 1176, 553, 200, 54, 10, 1

Offset: 1

Vladimir Kruchinin, Dec 14 2011

From Philippe Deléham, Feb 16 2014: (Start)
As a Riordan array, this is (1/(1 - x - x^2 - x^3 - x^4), x/(1 - x - x^2 - x^3 - x^4)).
T(n,0) = A000078(n+3); T(n+1,1) = A118898(n+4).
Row sums are A103142(n).
Diagonal sums are A077926(n)*(-1)^n.
Tetranacci convolution triangle. (End)

			Triangle begins:
   1;
   1,  1;
   2,  2,  1;
   4,  5,  3,  1;
   8, 12,  9,  4,  1;
  15, 28, 25, 14,  5,  1;
  29, 62, 66, 44, 20,  6,  1;

Similar sequences : A037027 (Fibonacci convolution triangle), A104580 (tribonacci convolution triangle). - Philippe Deléham, Feb 16 2014

T(n,m):=if n=m then 1 else sum(sum((-1)^i*binomial(k,k-i)*binomial(n-m-4*i-1,k-1),i,0,(n-m-k)/4)*binomial(k+m-1,m-1),k,1,n-m);

T(n,m) = Sum_{k=1..n-m} (Sum_{i=0..floor((n-m-k)/4)} (-1)^i*binomial(k,k-i)*binomial(n-m-4*i-1,k-1))*binomial(k+m-1,m-1), n > m, T(n,n)=1.
T(n,k) = T(n-1,k) + T(n-1,k-1) + T(n-2,k) + T(n-3,k) + T(n-4,k), T(0,0) = 1, T(n,k) = 0 if k < 0 or if k > n. - Philippe Deléham, Feb 16 2014
G.f. for column m: (x/(1 - x - x^2 - x^3 - x^4))^m. - Jason Yuen, Feb 17 2025

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A103141 Riordan array (1/(1-x), x*(1 + x + x^2 + x^3)/(1-x)).

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A141448 Generalized Pell numbers P(n,5,5).

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A190139 a(n) = 2*a(n-1) + a(n-2) + a(n-3) + a(n-4), a(-2)=0, a(-1)=0, a(0)=1, a(1)=1.

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A202193 Triangle read by rows: T(n,m) = coefficient of x^n in expansion of (x/(1 - x - x^2 - x^3 - x^4))^m.

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