A071675 -id:A071675 - cp's OEIS frontend

A103779 Series reversion of y + y^2 + y^3.

0, 1, -1, 1, 0, -4, 14, -30, 33, 55, -429, 1365, -2652, 1428, 12920, -64600, 178296, -277932, -152950, 2770350, -10785390, 25312650, -26053020, -84847620, 576753450, -1856900682, 3566658438, -843350102, -24973594296, 117328602840, -317641049880, 455822225496

Offset: 0

Paul Barry, Feb 15 2005

Second column of A103778 (inverse of trinomial triangle A071675).

			G.f.: A(x) = x - x^2 + x^3 - 4*x^5 + 14*x^6 - 30*x^7 + 33*x^8 + 55*x^9 - 429*x^10 + 1365*x^11 - 2652*x^12 + 1428*x^13 + 12920*x^14 + ... where A(x + x^2 + x^3) = x.

R. J. Mathar, Table of n, a(n) for n = 0..103
Elżbieta Liszewska and Wojciech Młotkowski, Some relatives of the Catalan sequence, arXiv:1907.10725 [math.CO], 2019.

Cf. A350383.

CoefficientList[ InverseSeries[ Series[y + y^2 + y^3, {y, 0, 28}], x], x] (* Robert G. Wilson v *)

a(n):=if n=1 then 1 else -sum(sum(binomial(j,n-k-j) *binomial(k,j),j,0,k)*a(k),k,1,n-1); /* Vladimir Kruchinin, Apr 08 2011 */

a(n):=if n=1 then 1 else 1/n*sum(binomial(k,n-1-k)*(-1)^k *binomial(n+k-1,n-1),k,1,n-1); /* Vladimir Kruchinin, May 12 2012 */

Vec(serreverse(x*(1+x+x^2)+O(x^66))) /* Joerg Arndt, Aug 19 2012 */

/* G.f. A(x) satisfies: A(x)^2 = A( x^2 - 2*x*A(x)^2 ) */
{a(n) = my(A=x+x^2, X=x+x*O(x^n)); for(i=1, n, A = subst(A, x, x^2 - 2*X*A^2)^(1/2) ); polcoeff(A, n)}
for(n=1, 40, print1(a(n), ", ")) \\ Paul D. Hanna, Apr 17 2016

G.f.: -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.
a(0) = 0, a(1) = 1, and, for n >= 2, a(n) = -Sum_{k = 1..n-1} a(k) * Sum_{j = 0..k} C(j,n-k-j) * C(k,j). - Vladimir Kruchinin, Apr 08 2011
a(n) = (1/n)*Sum_{k = 1..n-1} (-1)^k * C(k,n-1-k) * C(n+k-1,n-1), with a(1) = 1. - Vladimir Kruchinin, May 12 2012
D-finite with recurrence 3*n*(n-1)*a(n) +7*(n-1)*(2*n-3)*a(n-1) +3*(3*n-5)*(3*n-7)*a(n-2)=0. - R. J. Mathar, Oct 06 2012
G.f. A(x) satisfies: A(x)^2 = A( x^2 - 2*x*A(x)^2 ). - Paul D. Hanna, Apr 17 2016
From Paul D. Hanna, Sep 06 2022: (Start)
G.f. A(x) satisfies:
A(x)^5 = A( x^5 - 5*x*(1+x)^2*A(x)^5 ), and
A(x)^5 = ( x^5 - 5*x*(1+x)^2*A(x)^5 ) * (1 - A(x)^5) / (1 - A(x)^15). (End)
A(x) = x*exp(Sum_{n >= 1} A350383(n)*x^n/n). - Peter Bala, Mar 11 2025

A213889 Triangle of coefficients of representations of columns of A213745 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, 1, 5, 10, 10, 5, 1, 0, 0, 6, 15, 20, 15, 6, 1, 0, 0, 5, 21, 35, 35, 21, 7, 1, 0, 0, 4, 25, 56, 70, 56, 28, 8, 1, 0, 0, 3, 27, 80, 126, 126, 84, 36, 9, 1

Offset: 0

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

This array is the fifth array in the sequence of arrays A026729, A071675, A213887, A213888,..., 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 A213745. For example, s_1(n)=binomial(n,1)=n is the first column of A213745 for n>1, s_2(n)=binomial(n,1)+binomial(n,2)is the second column of A213745 for n>1, etc. In particular (see comment in A213745), in cases k=8,9 s_k(n) is A063417(n+2), A063418(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....1....5...10...10....5....1
.7..|..0....0....6...15...20...15....6....1
.8..|..0....0....5...21...35...35...21....7....1
.9..|..0....0....4...25...56...70...56...28....8....1

Cf. A026729, A071675, A078803 (parts <=3), A213887 (parts <=4), A213888 (parts <=5).

Essentially the same as A061676.

pts := 6; # A213889 and A061676
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

A005719 Quadrinomial coefficients.

Original entry on oeis.org

2, 12, 40, 101, 216, 413, 728, 1206, 1902, 2882, 4224, 6019, 8372, 11403, 15248, 20060, 26010, 33288, 42104, 52689, 65296, 80201, 97704, 118130, 141830, 169182, 200592, 236495, 277356, 323671, 375968, 434808, 500786, 574532, 656712, 748029, 849224, 961077

Offset: 2

N. J. A. Sloane

nonn

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).

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

a(n)= A008287(n, 5), n >= 2 (sixth column of quadrinomial coefficients).

A005719:=(2-2*z**2+z**3)/(z-1)**6; [Conjectured by Simon Plouffe in his 1992 dissertation.]

a(n)= binomial(n, 2)*(n^3+11*n^2+46*n-24)/60, n >= 2.
G.f.: (x^2)*(2-2*x^2+x^3)/(1-x)^6. (numerator polynomial is N4(5, x) from A063421.)
a(n) = 2*binomial(n,2) + 6*binomial(n,3) + 4*binomial(n,4) + binomial(n,5) (see comment in A071675). - Vladimir Shevelev and Peter J. C. Moses, Jun 22 2012

A005720 Quadrinomial coefficients.

Original entry on oeis.org

1, 10, 44, 135, 336, 728, 1428, 2598, 4455, 7282, 11440, 17381, 25662, 36960, 52088, 72012, 97869, 130986, 172900, 225379, 290444, 370392, 467820, 585650, 727155, 895986, 1096200, 1332289, 1609210, 1932416, 2307888, 2742168, 3242393, 3816330, 4472412, 5219775

Offset: 2

N. J. A. Sloane

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).

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 (7,-21,35,-35,21,-7,1).

a(n)= A008287(n, 6), n >= 2 (seventh column of quadrinomial coefficients).

A005720:=-(1+3*z-5*z**2+2*z**3)/(z-1)**7; [Conjectured by Simon Plouffe in his 1992 dissertation.]

LinearRecurrence[{7,-21,35,-35,21,-7,1},{1,10,44,135,336,728,1428},40] (* or *) Table[Binomial[n+1,3] (n^3+15n^2+86n-120)/120,{n,2,41}] (* Harvey P. Dale, Jun 23 2011 *)

a(n)=(n^6 + 15*n^5 + 85*n^4 - 135*n^3 - 86*n^2 + 120*n)/720 \\ Charles R Greathouse IV, Jun 23 2011

a(n)= binomial(n+1, 3)*(n^3+15*n^2+86*n-120)/120, n >= 2.
G.f.: (x^2)*(1+3*x-5*x^2+2*x^3)/(1-x)^7. (numerator polynomial is N4(6, x) from A063421).
a(0)=1, a(1)=10, a(2)=44, a(3)=135, a(4)=336, a(5)=728, a(6)=1428, 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). - Harvey P. Dale, Jun 23 2011
a(n) = binomial(n,2) + 7*binomial(n,3) + 10*binomial(n,4) + 5*binomial(n,5) + binomial(n,6) (see comment in A071675). - Vladimir Shevelev and Peter J. C. Moses, Jun 22 2012

A064055 Ninth column of quadrinomial coefficients.

Original entry on oeis.org

3, 31, 155, 546, 1554, 3823, 8451, 17205, 32802, 59268, 102388, 170261, 273975, 428418, 653242, 973998, 1423461, 2043165, 2885169, 4014076, 5509328, 7467801, 10006725, 13266955, 17416620, 22655178, 29217906

Offset: 0

Wolfdieter Lang, Aug 29 2001

A001919 (eighth column).

Table[3Binomial[n+3,3]+19Binomial[n+3,4]+30Binomial[n+3,5]+21 Binomial[n+3,6]+ 7 Binomial[n+3,7]+ Binomial[n+3,8],{n,0,30}] (* Harvey P. Dale, Apr 30 2022 *)

a(n)= A008287(n+3, 8)= binomial(n+3, 3)*(n^5+46*n^4+875*n^3+7118*n^2+23880*n+20160)/(8!/3!), n >= 0.
G.f.: (3+4*x-16*x^2+15*x^3-6*x^4+x^5 )/(1-x)^9, numerator polynomial is N4(8, x) from the array A063421.
a(n) = 3*C(n+3,3) + 19*C(n+3,4) + 30*C(n+3,5) + 21*C(n+3,6) + 7*C(n+3,7) + C(n+3,8) (see comment in A071675). - Vladimir Shevelev and Peter J. C. Moses, Jun 22 2012

A077828 Expansion of 1/(1-3x-3x^2-3*x^3).

Original entry on oeis.org

1, 3, 12, 48, 189, 747, 2952, 11664, 46089, 182115, 719604, 2843424, 11235429, 44395371, 175422672, 693160416, 2738935377, 10822555395, 42763953564, 168976333008, 667688525901, 2638286437419, 10424853888984, 41192486556912, 162766880649945, 643152663287523

Offset: 0

N. J. A. Sloane, Nov 17 2002

Michael De Vlieger, Table of n, a(n) for n = 0..1675
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 (3,3,3).

Partial sums of S(n, x), for x=1...12, A021823, A000217, A027941, A061278, A089817, A053142, A092521, A076765, A092420, A097784, A097826-7.

Cf. A071675.

CoefficientList[Series[1/(1-3x-3x^2-3x^3),{x,0,30}],x] (* or *) LinearRecurrence[ {3,3,3},{1,3,12},30] (* Harvey P. Dale, Dec 25 2018 *)

Vec(1/(1-3*x-3*x^2-3*x^3)+O(x^99)) \\ Charles R Greathouse IV, Sep 27 2012

a(n) = sum{k=0..n, T(n-k, k)3^(n-k)}, T(n, k) = trinomial coefficients (A027907). - Paul Barry, Feb 15 2005

a(n) = sum{k=0..n, sum{i=0..floor((n-k)/2), C(n-k-i, i)C(k, n-k-i)}*3^k}. - Paul Barry, Apr 26 2005

A103770 A weighted tribonacci sequence, (1,3,9).

Original entry on oeis.org

1, 1, 4, 16, 37, 121, 376, 1072, 3289, 9889, 29404, 88672, 265885, 796537, 2392240, 7174816, 21520369, 64574977, 193709428, 581117680, 1743420757, 5230158649, 15690480040, 47071742800, 141214610761, 423644159521, 1270933677004

Offset: 0

Paul Barry, Feb 15 2005

The weighted tribonacci (1,r,r^2) with g.f. 1/(1 - x - r*x^2 - r^2*x^3) has general term Sum_{k=0..n} T(n-k,k)r^k.
Correspondence: a(n) = b(n+2)*3^n, where b(n) is the sequence of the arithmetic means of the previous three terms defined by b(n) = (1/3)*(b(n-1) + b(n-2) + b(n-3)) with initial values b(0)=0, b(1)=0, b(2)=1; the g.f. for b(n) is B(x) := x^2/(1-(x^1+x^2+x^3)/3), so the g.f. A(x) for a(n) satisfies A(x) = B(3*x)/(3*x)^2. Because b(n) converges to the limit lim_{x->1} (1-x)*B(x) = (1/6)*(b(0) + 2*b(1) + 3*b(2)) = 1/2, it follows that a(n)/3^n also converges to 1/2. This correspondence is valid in general (with necessary changes) for weighted sequences of order (1, p, p^2, p^3, p^4, ..., p^(p-1)) with integer p > 0. Forming such sequences c(n) := c(n-1) + p^1*c(n-2) + ... + p^(p-1)*c(n-p) the limit of c(n)/p^n is 2/(p+1) (see also A001045). - Hieronymus Fischer, Feb 04 2006
a(n)/3^n equals the probability that n will occur as a partial sum in a randomly-generated infinite sequence of 1s, 2s and 3s. The limiting ratio is 1/2. - Bob Selcoe, Jul 05 2013
Number of compositions of n into one sort of 1's, three sorts of 2's, and nine sorts of 3's. - Joerg Arndt, Jul 06 2013
Using the Markov Chain {{0, 1, 0}, {0, 0, 1}, {1/3, 1/3, 1/3}} and raising it to the n-th power can generate this sequence when looking at the element in the third row and third column and reading the numerator. - Robert P. P. McKone, May 25 2021

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 (1,3,9).

Cf. A000073, A102001.
Cf. A071675, A027907.
Cf. A121262, A088137.

LinearRecurrence[{1, 3, 9}, {1, 1, 4}, {1, 27}] (* Robert P. P. McKone, May 25 2021 *)

G.f.: 1/(1 - x - 3*x^2 - 9*x^3).
a(n) = Sum_{k=0..n} T(n-k, k)*3^k, T(n, k) = trinomial coefficients (A027907).
a(n) = Sum_{k=0..n} 3^(n-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)/3^n converges to 1/2. - Hieronymus Fischer, Feb 02 2006
a(n) = a(n-1) + 3*a(n-2) + 9*a(n-3), n >= 3; a(0)=1, a(1)=1, a(2)=4. - Hieronymus Fischer, Feb 04 2006
a(n) = 3^n + b(n) + b(n-1), with b(n) = (-1)^A121262(n+1)*A088137(n+1). - Ralf Stephan, May 20 2007

A071676 Array read by antidiagonals of signed variant of trinomial coefficients with T(n,k)=T(n-1,k)+T(n-1,k-1)-T(n-1,k-2) starting with T(0,0)=1.

Original entry on oeis.org

1, 0, 1, 0, 1, 1, 0, -1, 2, 1, 0, 0, -1, 3, 1, 0, 0, -2, 0, 4, 1, 0, 0, 1, -5, 2, 5, 1, 0, 0, 0, 0, -8, 5, 6, 1, 0, 0, 0, 3, -5, -10, 9, 7, 1, 0, 0, 0, -1, 8, -15, -10, 14, 8, 1, 0, 0, 0, 0, 2, 11, -30, -7, 20, 9, 1, 0, 0, 0, 0, -4, 15, 6, -49, 0, 27, 10, 1, 0, 0, 0, 0, 1, -10, 41, -14, -70, 12, 35, 11, 1, 0, 0, 0, 0, 0, -5, -6, 77, -56, -90, 30, 44

Offset: 0

Henry Bottomley, May 30 2002

Each column is eventually positive, i.e. for each k there is a number j(k) such that T(n,k) is positive for all n>=j(k). Despite this, each row sum is 1.

			Rows start:
1, 0,  0,  0,  0, ...;
1, 1, -1,  0,  0, 0, ...;
1, 2, -1, -2,  1, 0,  0, ...;
1, 3,  0, -5,  0, 3, -1,  0, ...;
1, 4,  2, -8, -5, 8,  2, -4, 1, ...;
etc...

Cf. A071675.

cp's OEIS Frontend

A103779 Series reversion of y + y^2 + y^3.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

Maxima

Maxima

PARI

PARI

Formula

A213889 Triangle of coefficients of representations of columns of A213745 in binomial basis.

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

Maple

A005719 Quadrinomial coefficients.

Views

Author

Keywords

References

Links

Crossrefs

Programs

Maple

Formula

A005720 Quadrinomial coefficients.

Views

Author

Keywords

References

Links

Crossrefs

Programs

Maple

Mathematica

PARI

Formula

A064055 Ninth column of quadrinomial coefficients.

Views

Author

Keywords

Crossrefs

Programs

Mathematica

Formula

A077828 Expansion of 1/(1-3*x-3*x^2-3*x^3).

Views

Author

Keywords

Links

Crossrefs

Programs

Mathematica

PARI

Formula

A103770 A weighted tribonacci sequence, (1,3,9).

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Mathematica

Formula

A071676 Array read by antidiagonals of signed variant of trinomial coefficients with T(n,k)=T(n-1,k)+T(n-1,k-1)-T(n-1,k-2) starting with T(0,0)=1.

Views

Author

Keywords

Comments

A077828 Expansion of 1/(1-3x-3x^2-3*x^3).