A055275 -id:A055275 - cp's OEIS frontend

A167374 Triangle, read by rows, given by [ -1,1,0,0,0,0,0,0,0,...] DELTA [1,0,0,0,0,0,0,0,0,...] where DELTA is the operator defined in A084938.

1, -1, 1, 0, -1, 1, 0, 0, -1, 1, 0, 0, 0, -1, 1, 0, 0, 0, 0, -1, 1, 0, 0, 0, 0, 0, -1, 1, 0, 0, 0, 0, 0, 0, -1, 1, 0, 0, 0, 0, 0, 0, 0, -1, 1, 0, 0, 0, 0, 0, 0, 0, 0, -1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, -1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, -1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, -1, 1

Offset: 0

Philippe Deléham, Nov 02 2009

Riordan array (1-x,1) read by rows; Riordan inverse is (1/(1-x),1). Columns have g.f. (1-x)x^k. Diagonal sums are A033999. Unsigned version in A097806.
Table T(n,k) read by antidiagonals. T(n,1) = 1, T(n,2) = -1, T(n,k) = 0, k > 2. - Boris Putievskiy, Jan 17 2013
Finite difference operator (pair difference): left multiplication by T of a sequence arranged as a column vector gives a running forward difference, a(k+1)-a(k), or first finite difference (modulo sign), of the elements of the sequence. T^n gives the n-th finite difference (mod sign). T is the inverse of the summation matrix A000012 (regarded as lower triangular matrices). - Tom Copeland, Mar 26 2014

			Triangle begins:
   1;
  -1,  1;
   0, -1,  1;
   0,  0, -1,  1;
   0,  0,  0, -1,  1;
   0,  0,  0,  0, -1,  1; ...
Row number r (r>4) contains (r-2) times '0', then '-1' and '1'.
From _Boris Putievskiy_, Jan 17 2013: (Start)
The start of the sequence as a table:
  1  -1  0  0  0  0  0 ...
  1  -1  0  0  0  0  0 ...
  1  -1  0  0  0  0  0 ...
  1  -1  0  0  0  0  0 ...
  1  -1  0  0  0  0  0 ...
  1  -1  0  0  0  0  0 ...
  1  -1  0  0  0  0  0 ...
  ...
(End)

Boris Putievskiy, Transformations [of] Integer Sequences And Pairing Functions arXiv:1212.2732 [math.CO], 2012.

Cf. A000012, A007318, A029653, A097805, A118800, A130595, A156644.

A167374 := proc(n,k)
    if k> n or k < n-1 then
        0;
    elif k = n then
        1;
    else
        -1 ;
    end if;
end proc: # R. J. Mathar, Sep 07 2016

Table[PadLeft[{-1, 1}, n], {n, 13}] // Flatten (* or *)
MapIndexed[Take[#1, First@ #2] &, CoefficientList[Series[(1 - x)/(1 - x y), {x, 0, 12}], {x, y}]] // Flatten (* Michael De Vlieger, Nov 16 2016 *)
T[n_, k_] := If[ k<0 || k>n, 0, Boole[n==k] - Boole[n==k+1]]; (* Michael Somos, Oct 01 2022 *)

{T(n, k) = if( k<0 || k>n, 0, (n==k) - (n==k+1))}; /* Michael Somos, Oct 01 2022 */

Sum_{k, 0<=k<=n} T(n,k)*x^k = A000007(n), A011782(n), A025192(n), A002001(n), A005054(n), A052934(n), A055272(n), A055274(n), A055275(n), A055268(n), A055276(n) for x = 1,2,3,4,5,6,7,8,9,10,11 respectively .
From Boris Putievskiy, Jan 17 2013: (Start)
a(n) = floor((A002260(n)+2)/(A003056(n)+2))*(-1)^(A002260(n)+A003056(n)+1),  n>0.
a(n) = floor((i+2)/(t+2))*(-1)^(i+t+1), n > 0, where
i = n - t*(t+1)/2,
t = floor((-1 + sqrt(8*n-7))/2). (End)
T*A000012 = Identity matrix. T*A007318 = A097805. T*(A007318)^(-1)= signed A029653. - Tom Copeland, Mar 26 2014
G.f.: (1-x)/(1-x*y). - R. J. Mathar, Aug 11 2015
T = A130595*A156644 = M*T^(-1)*M = M*A000012*M, where M(n,k) = (-1)^n A130595(n,k). Note that M = M^(-1). Cf. A118800 and A097805. - Tom Copeland, Nov 15 2016

A200139 Triangle T(n,k), read by rows, given by (1,1,0,0,0,0,0,0,0,...) DELTA (1,0,0,0,0,0,0,0,0,...) where DELTA is the operator defined in A084938.

Original entry on oeis.org

1, 1, 1, 2, 3, 1, 4, 8, 5, 1, 8, 20, 18, 7, 1, 16, 48, 56, 32, 9, 1, 32, 112, 160, 120, 50, 11, 1, 64, 256, 432, 400, 220, 72, 13, 1, 128, 576, 1120, 1232, 840, 364, 98, 15, 1, 256, 1280, 2816, 3584, 2912, 1568, 560, 128, 17, 1, 512, 2816, 6912, 9984, 9408, 6048, 2688, 816, 162, 19, 1

Offset: 0

Philippe Deléham, Nov 13 2011

Riordan array ((1-x)/(1-2x),x/(1-2x)).
Product A097805*A007318 as infinite lower triangular arrays.
Product A193723*A130595 as infinite lower triangular arrays.
T(n,k) is the number of ways to place n unlabeled objects into any number of labeled bins (with at least one object in each bin) and then designate k of the bins. - Geoffrey Critzer, Nov 18 2012
Apparently, rows of this array are unsigned diagonals of A028297. - Tom Copeland, Oct 11 2014
Unsigned A118800, so my conjecture above is true. - Tom Copeland, Nov 14 2016

			Triangle begins:
   1
   1,   1
   2,   3,   1
   4,   8,   5,   1
   8,  20,  18,   7,   1
  16,  48,  56,  32,   9,   1
  32, 112, 160, 120,  50,  11,   1

Cf. A118800 (signed version), A081277, A039991, A001333 (antidiagonal sums), A025192 (row sums); diagonals: A000012, A005408, A001105, A002492, A072819l; columns: A011782, A001792, A001793, A001794, A006974, A006975, A006976.

Cf. A007318, A028297, A059260, A097805, A118801, A130595.

nn=15;f[list_]:=Select[list,#>0&];Map[f,CoefficientList[Series[(1-x)/(1-2x-y x) ,{x,0,nn}],{x,y}]]//Grid  (* Geoffrey Critzer, Nov 18 2012 *)

T(n,k) = 2*T(n-1,k)+T(n-1,k-1) with T(0,0)=T(1,0)=T(1,1)=1 and T(n,k)=0 for k<0 or for n

T(n,k) = A011782(n-k)*A135226(n,k) = 2^(n-k)*(binomial(n,k)+binomial(n-1,k-1))/2.

Sum_{k, 0<=k<=n} T(n,k)*x^k = A000007(n), A011782(n), A025192(n), A002001(n), A005054(n), A052934(n), A055272(n), A055274(n), A055275(n), A052268(n), A055276(n), A196731(n) for n=-1,0,1,2,3,4,5,6,7,8,9,10 respectively.

G.f.: (1-x)/(1-(2+y)*x).

T(n,k) = Sum_j>=0 T(n-1-j,k-1)*2^j.

T = A007318*A059260, so the row polynomials of this entry are given umbrally by p_n(x) = (1 + q.(x))^n, where q_n(x) are the row polynomials of A059260 and (q.(x))^k = q_k(x). Consequently, the e.g.f. is exp[tp.(x)] = exp[t(1+q.(x))] = e^t exp(tq.(x)) = [1 + (x+1)e^((x+2)t)]/(x+2), and p_n(x) = (x+1)(x+2)^(n-1) for n > 0. - Tom Copeland, Nov 15 2016

T^(-1) = A130595*(padded A130595), differently signed A118801. Cf. A097805. - Tom Copeland, Nov 17 2016

The n-th row polynomial in descending powers of x is the n-th Taylor polynomial of the rational function (1 + x)/(1 + 2*x) * (1 + 2*x)^n about 0. For example, for n = 4, (1 + x)/(1 + 2*x) * (1 + 2*x)^4 = (8*x^4 + 20*x*3 + 18*x^2 + 7*x + 1) + O(x^5). - Peter Bala, Feb 24 2018

A193723 Mirror of the fusion triangle A193722.

Original entry on oeis.org

1, 2, 1, 6, 5, 1, 18, 21, 8, 1, 54, 81, 45, 11, 1, 162, 297, 216, 78, 14, 1, 486, 1053, 945, 450, 120, 17, 1, 1458, 3645, 3888, 2295, 810, 171, 20, 1, 4374, 12393, 15309, 10773, 4725, 1323, 231, 23, 1, 13122, 41553, 58320, 47628, 24948, 8694, 2016, 300, 26, 1

Offset: 0

Clark Kimberling, Aug 04 2011

A193723 is obtained by reversing the rows of the triangle A193722.
Triangle T(n,k), read by rows, given by [2,1,0,0,0,0,0,0,0,...] DELTA [1,0,0,0,0,0,0,0,...] where DELTA is the operator defined in A084938. - Philippe Deléham, Oct 04 2011
From Philippe Deléham, Nov 14 2011: (Start)
Riordan array ((1-x)/(1-3x), x/(1-3x)).
Product A200139*A007318 as infinite lower triangular arrays. (End)

			First six rows:
    1;
    2,   1;
    6,   5,   1;
   18,  21,   8,   1;
   54,  81,  45,  11,   1;
  162, 297, 216,  78,  14,   1;

Cf. A084938, A193722, A052924 (antidiagonal sums), Diagonals: A000012, A016789, A081266, Columns: A025192, A081038.

z = 9; a = 1; b = 1; c = 1; d = 2;
p[n_, x_] := (a*x + b)^n ; q[n_, x_] := (c*x + d)^n
t[n_, k_] := Coefficient[p[n, x], x^k]; t[n_, 0] := p[n, x] /. x -> 0;
w[n_, x_] := Sum[t[n, k]*q[n + 1 - k, x], {k, 0, n}]; w[-1, x_] := 1
g[n_] := CoefficientList[w[n, x], {x}]
TableForm[Table[Reverse[g[n]], {n, -1, z}]]
Flatten[Table[Reverse[g[n]], {n, -1, z}]] (* A193722 *)
TableForm[Table[g[n], {n, -1, z}]]
Flatten[Table[g[n], {n, -1, z}]] (* A193723 *)

Write w(n,k) for the triangle at A193722.  The triangle at A193723 is then given by w(n,n-k).
T(n,k) = T(n-1,k-1) + 3*T(n-1,k) with T(0,0)=T(1,1)=1 and T(1,0)=2. - Philippe Deléham, Oct 05 2011
From Philippe Deléham, Nov 14 2011: (Start)
Sum_{k=0..n} T(n,k)*x^k = A000007(n), A011782(n), A025192(n), A002001(n), A005054(n), A052934(n), A055272(n), A055274(n), A055275(n), A052268(n), A055276(n), A196731(n) for x=-2,-1,0,1,2,3,4,5,6,7,8,9 respectively.
T(n,k) = Sum_{j>=0} T(n-1-j,k-1)*3^j.
G.f.: (1-x)/(1-(3+y)*x). (End)

A085388 First differences of n^k.

Original entry on oeis.org

1, 1, 0, 1, 1, 0, 1, 2, 2, 0, 1, 3, 6, 4, 0, 1, 4, 12, 18, 8, 0, 1, 5, 20, 48, 54, 16, 0, 1, 6, 30, 100, 192, 162, 32, 0, 1, 7, 42, 180, 500, 768, 486, 64, 0, 1, 8, 56, 294, 1080, 2500, 3072, 1458, 128, 0, 1, 9, 72, 448, 2058, 6480, 12500, 12288, 4374, 256, 0, 1, 10, 90, 648

Offset: 1

Paul Barry, Jun 30 2003

T(n,k) is the number of k-digit numbers in base n; n,k >= 2. - Mohammed Yaseen, Nov 11 2022

			Rows begin
  1,   0,   0,   0,   0, ...
  1,   1,   2,   4,   8, ...
  1,   2,   6,  18,  54, ...
  1,   3,  12,  48, 192, ...
  1,   4,  20, 100, 500, ...

Rows include A011782, A025192, A002001, A005054, A052934, A055272, A055274, A055275, A052268, A055276.
Diagonals include A053506, A085389, A085390.
Row-wise binomial transform is A083064.

T(n,k) = (n-1)*n^(k-1) + 0^k/n. - Corrected by Mohammed Yaseen, Nov 11 2022

T(n,0) = 1; T(n,k) = n^k - n^(k-1) for k >= 1. - Mohammed Yaseen, Nov 11 2022

Offset corrected by Mohammed Yaseen, Nov 11 2022

A270369 Expansion of g.f. (1-7x)/(1-9x).

Original entry on oeis.org

1, 2, 18, 162, 1458, 13122, 118098, 1062882, 9565938, 86093442, 774840978, 6973568802, 62762119218, 564859072962, 5083731656658, 45753584909922, 411782264189298, 3706040377703682, 33354363399333138, 300189270593998242, 2701703435345984178, 24315330918113857602, 218837978263024718418

Offset: 0

Colin Barker, Mar 18 2016

Colin Barker, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (9).

Cf. A001019 (powers of 9), A054879 (partial sums), A132025.

Cf. similar sequences with g.f. (1-k*x)/(1-9*x) and k=0..8: A001019 (k=0; k=8 gives two initial 1's ), A055275 (k=1), A270472 (k=2), A092810 (k=3), A067403 (k=4), A270473 (k=5), A102518 (k=6), this sequence (k=7).

CoefficientList[Series[(1-7x)/(1-9x),{x,0,20}],x] (* or *) Join[ {1}, NestList[9#&,2,20]] (* Harvey P. Dale, Oct 15 2017 *)

PARI
```
Vec((1-7*x)/(1-9*x) + O(x^30))
```

G.f.: (1-7*x)/(1-9*x).
a(n) = 9*a(n-1) for n>1.
a(n) = 2*9^(n-1) for n>0.
From Amiram Eldar, May 08 2023: (Start)
Sum_{n>=0} 1/a(n) = 25/16.
Sum_{n>=0} (-1)^n/a(n) = 11/20.
Product_{n>=1} (1 - 1/a(n)) = A132025. (End)
E.g.f.: (2*exp(9*x) + 7)/9. - Elmo R. Oliveira, Mar 25 2025

A096977 a(n) = 4a(n-1) + 3a(n-2) - 14a(n-3) + 8a(n-4).

Original entry on oeis.org

0, 1, 2, 11, 36, 157, 598, 2447, 9672, 38913, 155194, 621683, 2484908, 9943269, 39765790, 159077719, 636281744, 2545185225, 10180624386, 40722730555, 162890456180, 651562756781, 2606249162982, 10425000380191, 41699994064216

Offset: 0

Paul Barry, Jul 17 2004

Original name was: A Jacobsthal summation.

The convolution of A024000 and A003683. Inverse binomial transform is A055275, with interpolated zeros.

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (4,3,-14,8).

Cf. A001654.

[4*4^n/27-4*(-2)^n/27+n/9: n in [0..30]]; // Vincenzo Librandi, Jul 01 2011

LinearRecurrence[{4,3,-14,8},{0,1,2,11},30] (* Harvey P. Dale, Jul 01 2015 *)

a(n)=(4*4^n-4*(-2)^n+3*n)/27 \\ Charles R Greathouse IV, Jul 01 2011

G.f.: x*(1-2*x)/((1-x)^2*(1+2*x)*(1-4*x)).
a(n) = 4*4^n/27 - 4*(-2)^n/27 + n/9.
a(n) = Sum_{k=0..n} A001045(k)^2.
a(n) = 4*a(n-1) + 3*a(n-2) - 14*a(n-3) + 8*a(n-4).

A270472 Expansion of g.f. (1-2x)/(1-9x).

Original entry on oeis.org

1, 7, 63, 567, 5103, 45927, 413343, 3720087, 33480783, 301327047, 2711943423, 24407490807, 219667417263, 1977006755367, 17793060798303, 160137547184727, 1441237924662543, 12971141321962887, 116740271897665983, 1050662447078993847, 9455962023710944623, 85103658213398501607

Offset: 0

Colin Barker, Mar 17 2016

Colin Barker, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (9).

Cf. A001019 (powers of 9), A005032, A187709 (partial sums).

Cf. A055275: (1-x)/(1-9*x); A092810: (1-3*x)/(1-9*x).

CoefficientList[Series[(1 - 2 x)/(1 - 9 x), {x, 0, 20}], x] (* Michael De Vlieger, Mar 18 2016 *)

PARI
```
Vec((1-2*x)/(1-9*x) + O(x^30))
```

G.f.: (1-2*x)/(1-9*x).
a(n) = 9*a(n-1) for n>1.
a(n) = 7*9^(n-1) for n>0.
a(n) = A005032(2*n-2). - R. J. Mathar, Jan 28 2025
E.g.f.: (7*exp(9*x) + 2)/9. - Elmo R. Oliveira, Mar 25 2025

A055995 a(n) = 64*9^(n-2), a(0)=1, a(1)=7.

Original entry on oeis.org

1, 7, 64, 576, 5184, 46656, 419904, 3779136, 34012224, 306110016, 2754990144, 24794911296, 223154201664, 2008387814976, 18075490334784, 162679413013056, 1464114717117504, 13177032454057536, 118593292086517824

Offset: 0

Barry E. Williams, Jun 04 2000

For n>=2, a(n) is equal to the number of functions f:{1,2,...,n}->{1,2,3,4,5,6,7,8,9} such that for fixed, different x_1, x_2 in {1,2,...,n} and fixed y_1, y_2 in {1,2,3,4,5,6,7,8,9} we have f(x_1)<>y_1 and f(x_2)<> y_2. - Milan Janjic, Apr 19 2007

a(n) is the number of generalized compositions of n when there are 8*i-1 different types of i, (i=1,2,...). - Milan Janjic, Aug 26 2010

A. H. Beiler, Recreations in the Theory of Numbers, Dover, N.Y., 1964, pp. 194-196.

Milan Janjic, Enumerative Formulas for Some Functions on Finite Sets
Index entries for linear recurrences with constant coefficients, signature (9)

Second differences of 9^n (A001019). Cf. A055275.

a(n) = 9a(n-1) + ((-1)^n)*C(2, 2-n).
G.f.: (1-x)^2/(1-9x).
a(n) = Sum_{k, 0<=k<=n} A201780(n,k)*7^k. - Philippe Deléham, Dec 05 2011

A117861 Number of palindromes of length n (in base 9).

Original entry on oeis.org

8, 8, 72, 72, 648, 648, 5832, 5832, 52488, 52488, 472392, 472392, 4251528, 4251528, 38263752, 38263752, 344373768, 344373768, 3099363912, 3099363912, 27894275208, 27894275208, 251048476872, 251048476872, 2259436291848, 2259436291848, 20334926626632

Offset: 1

Martin Renner, May 02 2006

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

Cf. A050683.

Cf. A055275.

Table[8 9^Floor[(n-1)/2],{n,50}] (* Harvey P. Dale, Oct 21 2011 *)

a(n) = 8*9^floor((n-1)/2).

G.f.: 8*x*(1+x)/(1-9*x^2). a(n) = 8*3^(n-2)*(2-(-1)^n). - Bruno Berselli, Oct 24 2011

More terms from Harvey P. Dale, Oct 21 2011

Showing 1-10 of 12 results. Next

cp's OEIS Frontend

A167374 Triangle, read by rows, given by [ -1,1,0,0,0,0,0,0,0,...] DELTA [1,0,0,0,0,0,0,0,0,...] where DELTA is the operator defined in A084938.

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A233155 T(n,k) = Number of n X k 0..2 arrays with no element x(i,j) adjacent to value 2-x(i,j) horizontally or antidiagonally.

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A200139 Triangle T(n,k), read by rows, given by (1,1,0,0,0,0,0,0,0,...) DELTA (1,0,0,0,0,0,0,0,0,...) where DELTA is the operator defined in A084938.

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A193723 Mirror of the fusion triangle A193722.

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A085388 First differences of n^k.

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A270369 Expansion of g.f. (1-7*x)/(1-9*x).

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A096977 a(n) = 4*a(n-1) + 3*a(n-2) - 14*a(n-3) + 8*a(n-4).

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A270472 Expansion of g.f. (1-2*x)/(1-9*x).

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A270369 Expansion of g.f. (1-7x)/(1-9x).

A096977 a(n) = 4a(n-1) + 3a(n-2) - 14a(n-3) + 8a(n-4).

A270472 Expansion of g.f. (1-2x)/(1-9x).