A140343 -id:A140343 - cp's OEIS frontend

A135356 Triangle T(n,k) read by rows: coefficients in the recurrence of sequences which equal their n-th differences.

2, 2, 0, 3, -3, 2, 4, -6, 4, 0, 5, -10, 10, -5, 2, 6, -15, 20, -15, 6, 0, 7, -21, 35, -35, 21, -7, 2, 8, -28, 56, -70, 56, -28, 8, 0, 9, -36, 84, -126, 126, -84, 36, -9, 2, 10, -45, 120, -210, 252, -210, 120, -45, 10, 0, 11, -55, 165, -330, 462, -462, 330, -165, 55, -11, 2

Offset: 1

Paul Curtz, Dec 08 2007, Mar 25 2008, Apr 28 2008

Sequences which equal their p-th differences obey recurrences a(n) = Sum_{s=1..p} T(p,s)*a(n-s).
This defines T(p,s) as essentially a signed version of a chopped Pascal triangle A014410, see A130785.
For cases like p=2, 4, 6, 8, 10, 12, 14, the denominator of the rational generating function of a(n) contains a factor 1-x; depending on the first terms in the sequences a(n), additional, simpler recurrences may exist if this cancels with a factor in the numerator. - R. J. Mathar, Jun 10 2008

			Triangle begins with row n=1:
  2;
  2,   0;
  3,  -3,  2;
  4,  -6,  4,    0;
  5, -10, 10,   -5,   2;
  6, -15, 20,  -15,   6,   0;
  7, -21, 35,  -35,  21,  -7,  2;
  8, -28, 56,  -70,  56, -28,  8,  0;
  9, -36, 84, -126, 126, -84, 36, -9, 2;

Alois P. Heinz, Rows n = 1..141, flattened

Related sequences: A000079 (n=1), A131577 (n=2), (A131708 , A130785, A131562, A057079) (n=3), (A000749, A038503, A009545, A038505) (n=4), A133476 (n=5), A140343 (n=6), A140342 (n=7).

Cf. A000225, A000984, A051049, A130785.

A135356:= func< n,k | k eq n select 1-(-1)^n else (-1)^(k+1)*Binomial(n,k) >;
[A135356(n,k): k in [1..n], n in [1..12]]; // G. C. Greubel, Apr 09 2023

T:= (p, s)->  `if`(p=s, 2*irem(p, 2), (-1)^(s+1) *binomial(p, s)):
seq(seq(T(p, s), s=1..p), p=1..11);  # Alois P. Heinz, Aug 26 2011

T[p_, s_]:= If[p==s, 2*Mod[s, 2], (-1)^(s+1)*Binomial[p, s]];
Table[T[p, s], {p, 12}, {s, p}]//Flatten (* Jean-François Alcover, Feb 19 2015, after Alois P. Heinz *)

def A135356(n,k):
    if (k==n): return 2*(n%2)
    else: return (-1)^(k+1)*binomial(n,k)
flatten([[A135356(n,k) for k in range(1,n+1)] for n in range(1,13)]) # G. C. Greubel, Apr 09 2023

T(n,k) = (-1)^(k+1)*A007318(n, k). T(n,n) = 1 - (-1)^n.
Sum_{k=1..n} T(n, k) = 2.
From G. C. Greubel, Apr 09 2023: (Start)
Sum_{k=1..n} (-1)^(k-1)*T(n, k) = 2*A051049(n-1).
Sum_{k=1..n-1} T(n, k) = (1 + (-1)^n).
Sum_{k=1..n-1} (-1)^(k-1)*T(n, k) = A000225(n-1).
T(2*n, n) = (-1)^(n-1)*A000984(n), n >= 1. (End)

Edited by R. J. Mathar, Jun 10 2008

A106233 An inverse Catalan transform of A003462.

Original entry on oeis.org

0, 1, 3, 5, 5, 0, -14, -41, -81, -121, -121, 0, 364, 1093, 2187, 3281, 3281, 0, -9842, -29525, -59049, -88573, -88573, 0, 265720, 797161, 1594323, 2391485, 2391485, 0, -7174454, -21523361, -43046721, -64570081, -64570081, 0, 193710244, 581130733, 1162261467

Offset: 0

Paul Barry, Apr 26 2005

The g.f. is obtained from that of A003462 through the mapping g(x)->g(x(1-x)). A003462 may be retrieved through the mapping g(x)->g(xc(x)), where c(x) is the g.f. of A000108. Binomial transform of x(1+x)/(1+x^2+x^4).

The sequence is identical to its sixth differences. See A140344. - Paul Curtz, Nov 09 2012

			From _Paul Curtz_, Nov 09 2012: (Start)
The sequence and its higher-order differences (periodic after 6 rows):
   0,  1,  3,  5,  5,   0, -14, ...
   1,  2,  2,  0, -5, -14, -27, ...
   1,  0, -2, -5, -9, -13, -13, ...
  -1, -2, -3, -4, -4,   0,  13, ...   = -A134581(n+1)
  -1, -1, -1,  0,  4,  13,  27, ...
   0,  0,  1,  4,  9,  14,  14, ...   = A140343(n+2)
   0,  1,  3,  5,  5,   0, -14, ...
(End)

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (4,-7,6,-3).

Cf. A103368.

I:=[0,1,3,5]; [n le 4 select I[n] else 4*Self(n-1)-7*Self(n-2)+ 6*Self(n-3)-3*Self(n-4): n in [1..40]]; // Vincenzo Librandi, Dec 24 2018

LinearRecurrence[{4, -7, 6, -3}, {0, 1, 3, 5}, 35] (* Vincenzo Librandi, Dec 24 2018 *)

G.f.: x(1-x)/((1-x+x^2)*(1-3*x+3*x^2));
a(n) = Sum_{k=0..floor(n/2)} C(n-k, k)*(-1)^k*(3^(n-k)-1)/2.
a(n) = Sum_{k=0..n} A109466(n,k)*A003462(k). - Philippe Deléham, Oct 30 2008
a(n) = (1/2)*[A057083(n) - [1,1,0,0,-1,-1]6 ]. - _Ralf Stephan, Nov 15 2010
a(n) = 4*a(n-1) - 7*a(n-2) + 6*a(n-3) - 3*a(n-4) = A140343(n+2) - A140343(n+1). - Paul Curtz, Nov 09 2012
a(n) is the binomial transform of the sequence 0, 1, 1, -1, -1, 0, ... = A103368(n+5). - Paul Curtz, Nov 09 2012

A140344 Catalan triangle A009766 prepended by n zeros in its n-th row.

Original entry on oeis.org

1, 0, 1, 1, 0, 0, 1, 2, 2, 0, 0, 0, 1, 3, 5, 5, 0, 0, 0, 0, 1, 4, 9, 14, 14, 0, 0, 0, 0, 0, 1, 5, 14, 28, 42, 42, 0, 0, 0, 0, 0, 0, 1, 6, 20, 48, 90, 132, 132, 0, 0, 0, 0, 0, 0, 0, 1, 7, 27, 75, 165, 297, 429, 429, 0, 0, 0, 0, 0, 0, 0, 0, 1, 8, 35, 110, 275, 572, 1001, 1430, 1430

Offset: 0

Paul Curtz, May 29 2008

The triangle's n-th row is also related to recurrences for sequences f(n) which p-th differences, p=n+2: The denominator of the generating function contains a factor 1-2x in these cases.
This factor may be "lifted" either by looking at auxiliary sequences f(n+1)-2f(n) or by considering the corresponding "degenerate" shorter recurrences right away. In the case p=4, the recurrence is f(n)=4f(n-1)-6f(n-2)+4f(n-3) from the 4th row in A135356, the denominator in the g.f. is 1-4x+6x^2-4x^3=(1-2x)(1-2x+2x^2), which yields the degenerate recurrence f(n)=2f(n-1)-2f(n-2) from the 2nd factor and leaves the first three coefficients of 1/(1-2x+2x^2)=1+2x+2x^2+.. in row 2.
A000749 is an example which follows the recurrence but not the degenerate recurrence, but still A000749(n+1)-2A000749(n) = 0, 0, 1, 2, 2,.. starts with the 3 coefficients. A009545 follows both recurrences and starts with the three nonzero terms because there is only a power of x in the numerator of the g.f.
In the case p=5, the recurrence is f(n)=5f(n-1)-10f(n-2)+10f(n-3)-5f(n-4)+2f(n-5), the denominator in the g.f. is 1-5x+10x^2-10x^3+5x^4-2x^5= (1-2x)(1-3x+4x^2-2x^3+x^4), where 1/(1-3x+4x^2-2x^3+x^4) = 1+3x+5x^2+5x^3+... and the 4 coefficients populate row 3.
A049016 obeys the main recurrence but not the degenerate recurrence f(n)=3f(n-1)-4f(n-2)+2f(n-3)-f(n-4), yet A049016(n+1)-2A049016(n)=1, 3, 5, 5,.. starts with the 4 coefficients. A138112 obeys both recurrences and is constructed to start with the 4 coefficients themselves.
In the nomenclature of Foata and Han, this is the doubloon polynomial triangle d_{n,m}(0), up to index shifts. - R. J. Mathar, Jan 27 2011

			Triangle starts
1;
0,1,1;
0,0,1,2,2;
0,0,0,1,3,5,5;
0,0,0,0,1,4,9,14,14;

D. Foata, G.-N. Han, The doubloon polynomial triangle, Ramanujan J. 23 (2010), 107-126

Cf. A135356, A130020, A139687, A140343 (p=6), A140342 (p=7).

Table[Join[Array[0&, n], Table[Binomial[n+k, n]*(n-k+1)/(n+1), {k, 0, n}]], {n, 0, 8}] // Flatten (* Jean-François Alcover, Dec 16 2014 *)

Edited by R. J. Mathar, Jul 10 2008

A139687 Basis of degenerate cases of sequences identical to its p-th differences. Complement to A140344 which is based on natural Catalan's triangle. Triangle without first term (probable 1) on line.

Original entry on oeis.org

1, 1, 1, 2, 2, 1, 3, 5, 5, 1, 3, 6, 9, 9, 1, 4, 10, 19, 28, 28, 1, 4, 10, 20, 34, 48, 48, 1, 5, 15, 35, 69, 117, 165, 165, 1, 5, 15, 35, 70, 125, 200, 275, 275, 1, 6, 21, 56, 126, 251, 451, 726, 1001, 1001, 1, 6, 21, 56, 126, 252, 461, 780, 1209, 1638, 1638

Offset: 0

Paul Curtz, Jun 13 2008

Triangle from A140344:
(1;)
0, 1, 1;
0, 0, 1, 2, 2;
0, 0, 0, 1, 3, 5, 5; see A138112,
0, 0, 0, 0, 1, 4, 9, 14, 14; see A140343,
begins (without 0's) like a(n).

Cf. A135356, A140344.

First four rows of triangle from second row: 1, 1; 1, 2, 2; see A099087, 1, 3, 5, 5; 1, 3, 6, 9, 9; see A057083 which can be preceded with 3 leading 0's, are, as said, from natural Catalan's triangle A009766. Origin of a(n) explained later.

A140342 a(n)=5a(n-1)-11a(n-2)+13a(n-3)-9a(n-4)+3a(n-5)-a(n-6).

Original entry on oeis.org

0, 0, 0, 0, 0, 1, 5, 14, 28, 42, 42, 0, -131, -417, -924, -1652, -2380, -2380, 0, 7753, 25213, 56714, 102256, 147798, 147798, 0, -479779, -1557649, -3499720, -6305992, -9112264, -9112264, 0, 29587889, 96072133, 215873462, 388991876, 562110290, 562110290, 0

Offset: 0

Paul Curtz, May 29 2008

This is the main sequence representing the degenerate case of sequences which equal their seventh difference, where besides the generic a(n)=7a(n-1)-21(n-2)+35a(n-3)-35a(n-4)+21a(n-5)-7a(n-6)+2a(n-7), cf. A135356, there is also a shorter recurrence.

After the first four terms, every seventh term is zero. - Harvey P. Dale, Sep 20 2012

Harvey P. Dale, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (5, -11, 13, -9, 3, -1).

Cf. A135356, A140343.

LinearRecurrence[{5,-11,13,-9,3,-1},{0,0,0,0,0,1},40] (* Harvey P. Dale, Sep 20 2012 *)

O.g.f.: x^5/(1-5x+11x^2-13x^3+9x^4-3x^5+x^6). - R. J. Mathar, Jul 10 2008

Edited and extended by R. J. Mathar, Jul 10 2008

A134581 a(n) = 4a(n-1) - 7a(n-2) + 6a(n-3) - 3a(n-4), starting with 0, 1, 2, 3.

Original entry on oeis.org

0, 1, 2, 3, 4, 4, 0, -13, -40, -81, -122, -122, 0, 365, 1094, 2187, 3280, 3280, 0, -9841, -29524, -59049, -88574, -88574, 0, 265721, 797162, 1594323, 2391484, 2391484, 0, -7174453, -21523360, -43046721, -64570082, -64570082, 0

Offset: 0

Paul Curtz, Jan 23 2008

Index entries for linear recurrences with constant coefficients, signature (4,-7,6,-3).

LinearRecurrence[{4, -7, 6, -3}, {0, 1, 2, 3}, 50] (* Harvey P. Dale, Dec 06 2013 *)
a[ n_] := Nest[# + RotateRight @ #&, {0, -1, 0, 0, 0, 1}, n][[1]]; (* Michael Somos, Jan 18 2023 *)

G.f.: x*(1-2*x+2*x^2)/((1-x+x^2)*(1-3*x+3*x^2)). - Jaume Oliver Lafont, Aug 30 2009
a(n) = A140343(n+3) - 2*A140343(n+2) + 2*A140343(n+1). - R. J. Mathar, Nov 21 2012
From Peter Bala, Jul 24 2017: (Start)
a(6*n) = 0;
a(6*n+1) = ((-1)^n*3^(3*n) + 1)/2;
a(6*n+2) = ((-1)^n*3^(3*n+1) + 1)/2;
a(6*n+3) = (-1)^n*3^(3*n+1);
a(6*n+4) = a(6*n+5) = ((-1)^n*3^(3*n+2) - 1)/2.
The o.g.f. A(x) satisfies (1 - x)*A(x) = x*A(1 - x).
Logarithmic g.f.: (1/sqrt(3))*arctan(sqrt(3)*x*(1 - x)/(1 - 2*x)) = Sum_{n >= 1} a(n)*x^n/n.
Sum_{n >= 1} a(n)/(n*2^n) = Pi/(2*sqrt(3)). (End)
a(n) = (3^(n/2) * sin(Pi*n/6) + sin(Pi*n/3)) / sqrt(3). - Peter Luschny, Jul 24 2017
2*a(n) = A010892(n-1) + A057083(n-1). - R. J. Mathar, Oct 03 2021
a(n) = -26*a(n-6) + 27*a(n-12) for all n in Z. - Michael Somos, Jan 18 2023

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A135356 Triangle T(n,k) read by rows: coefficients in the recurrence of sequences which equal their n-th differences.

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A106233 An inverse Catalan transform of A003462.

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A140344 Catalan triangle A009766 prepended by n zeros in its n-th row.

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A139687 Basis of degenerate cases of sequences identical to its p-th differences. Complement to A140344 which is based on natural Catalan's triangle. Triangle without first term (probable 1) on line.

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A140342 a(n)=5a(n-1)-11a(n-2)+13a(n-3)-9a(n-4)+3a(n-5)-a(n-6).

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A134581 a(n) = 4*a(n-1) - 7*a(n-2) + 6*a(n-3) - 3*a(n-4), starting with 0, 1, 2, 3.

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A134581 a(n) = 4a(n-1) - 7a(n-2) + 6a(n-3) - 3a(n-4), starting with 0, 1, 2, 3.