A130785 -id:A130785 - 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

A157823 a(n) = A156591(n) + A156591(n+1).

Original entry on oeis.org

-5, -1, -2, -4, -8, -16, -32, -64, -128, -256, -512, -1024, -2048, -4096, -8192, -16384, -32768, -65536, -131072, -262144, -524288, -1048576, -2097152, -4194304, -8388608, -16777216, -33554432, -67108864, -134217728, -268435456, -536870912, -1073741824

Offset: 0

Paul Curtz, Mar 07 2009

A156591 = 2,-7,6,-8,4,-12,... a(n) is companion to A154589 = 4,-1,-2,-4,-8,.For this kind ,companion of sequence b(n) is first differences a(n), second differences being b(n). Well known case: A131577 and A011782. a(n)+b(n)=A000079 or -A000079. a(n)=A154570(n+2)-A154570(n) ,A154570 = 1,3,-4,2,-6,-2,-14,. See sequence(s) identical to its p-th differences (A130785,A130781,A024495,A000749,A138112(linked to Fibonacci),A139761).

Colin Barker, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (2).

Vec(-(9*x-5)/(2*x-1) + O(x^100)) \\ Colin Barker, Feb 03 2015

a(n) = 2*a(n-1) for n>1. G.f.: -(9*x-5) / (2*x-1). - Colin Barker, Feb 03 2015

Edited by Charles R Greathouse IV, Oct 11 2009

A141775 Binomial transform of (1, 2, 0, 1, 2, 0, 1, 2, 0, ...).

Original entry on oeis.org

1, 3, 5, 8, 15, 31, 64, 129, 257, 512, 1023, 2047, 4096, 8193, 16385, 32768, 65535, 131071, 262144, 524289, 1048577, 2097152, 4194303, 8388607, 16777216, 33554433, 67108865, 134217728, 268435455, 536870911, 1073741824, 2147483649, 4294967297, 8589934592, 17179869183

Offset: 0

Gary W. Adamson, Jul 03 2008

From Paul Curtz, Jun 15 2011: (Start)
A square array of a(n) and its higher order differences is defined by T(0,k) = a(k) and T(n,k) = T(n-1,k+1)-T(n-1,k):
1, 3, 5, 8, 15, 31,
2, 2, 3, 7, 16, 33,
0, 1, 4, 9, 17, 32, see A130785(n).
1, 3, 5, 8, 15, 31,
2, 2, 3, 7, 16, 33,
a(n) is identical to its third differences: T(n+3,k) = T(n,k).
The main diagonal is T(n,n) = 2^n. Subdiagonals are T(n,n-1) = A014551(n) and T(n,n-2) = A062510(n).
(End)

			a(4) = 8 = (1, 2, 0, 1) dot (1, 3, 3, 1) = (1 + 6 + 0 + 1).

Harvey P. Dale, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (3,-3,2).

Cf. A000079, A057079.

I:=[1,3,5]; [n le 3 select I[n] else 3*Self(n-1) - 3*Self(n-2) + 2*Self(n-3): n in [1..30]]; // G. C. Greubel, Jan 15 2018

LinearRecurrence[{3,-3,2},{1,3,5},40] (* Harvey P. Dale, May 29 2012 *)

x='x+O('x^30); Vec((x-1)*(1+x)/((2*x-1)*(x^2-x+1))) \\ G. C. Greubel, Jan 15 2018

From Paul Curtz, Jun 15 2011: (Start)
a(n) = 3*a(n-1) - 3*a(n-2) + 2*a(n-3).
a(n) = 2^n - A128834(n).
a(n) - 2a(n-1)= A057079(n+1).
a(n) + a(n+3) = 9*2^n.
a(n+6) - a(n) = 63*2^n.
a(n) = A130785(n) - A130785(n-1). (End)
G.f.: (x-1)*(1+x) / ( (2*x-1)*(x^2-x+1) ). - R. J. Mathar, Jun 22 2011
a(n) = 2^n + (2*sin((Pi*n)/3))/sqrt(3). - Colin Barker, Feb 10 2017

A158745 a(3n)=A130750(n). a(3n+1)=A130752(n). a(3n+2)=A130755(n).

Original entry on oeis.org

1, 2, 3, 3, 5, 4, 8, 9, 7, 17, 16, 15, 33, 31, 32, 64, 63, 65, 127, 128, 129, 255, 257, 256, 512, 513, 511, 1025, 1024, 1023, 2049, 2047, 2048, 4096, 4095, 4097, 8191, 8192, 8193, 16383, 16385, 16384, 32768, 32769, 32767, 65537, 65536, 65535, 131073, 131071, 131072, 262144

Offset: 0

Paul Curtz, Mar 25 2009

This mixes three sequences which are identical to their third differences.

Harvey P. Dale, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (0, 0, 3, 0, 0, -3, 0, 0, 2).

Cf. A138635, A140430, A130785, A140495.

LinearRecurrence[{0,0,3,0,0,-3,0,0,2},{1,2,3,3,5,4,8,9,7},60] (* Harvey P. Dale, Mar 12 2023 *)

a(3n)+a(3n+1)+a(3n+2)= A007283(n+1).
a(18n) = A130750(6n)= 2^(6n+1)-1.
a(n) = 3*a(n-3)-3*a(n-6)+2*a(n-9). G.f.: -(1+2*x+3*x^2-x^4-5*x^5+2*x^6+4*x^8)/((2*x^3-1)*(x^6-x^3+1)). - R. J. Mathar, Jan 23 2009

Edited and extended by R. J. Mathar, Apr 09 2009

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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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A157823 a(n) = A156591(n) + A156591(n+1).

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A141775 Binomial transform of (1, 2, 0, 1, 2, 0, 1, 2, 0, ...).

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A158745 a(3n)=A130750(n). a(3n+1)=A130752(n). a(3n+2)=A130755(n).

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