A046854 -id:A046854 - cp's OEIS frontend

A106180 Matrix inverse of number triangle A046854.

1, -1, 1, 0, -1, 1, 1, -1, -1, 1, 0, 2, -2, -1, 1, -2, 2, 3, -3, -1, 1, 0, -5, 5, 4, -4, -1, 1, 5, -5, -9, 9, 5, -5, -1, 1, 0, 14, -14, -14, 14, 6, -6, -1, 1, -14, 14, 28, -28, -20, 20, 7, -7, -1, 1, 0, -42, 42, 48, -48, -27, 27

Offset: 0

Paul Barry, Apr 24 2005

First column is A105523; second column is A106181.
Triangle T(n,k), 0 <= k <= n, read by rows given by [ -1, 1, -1, 1, -1, 1, -1, 1, -1, 1,...] DELTA [1, 0, 0, 0, 0, 0, 0, 0, 0, ...] where DELTA is the operator defined in A084938. - Philippe Deléham, Sep 29 2006
A124448*A007318 as infinite lower triangular matrices. - Philippe Deléham, Oct 16 2007

			Triangle begins
   1;
  -1,  1;
   0, -1,  1;
   1, -1, -1,  1;
   0,  2, -2, -1,  1;
  -2,  2,  3, -3, -1,  1;
   0, -5,  5,  4, -4, -1,  1;

Cf. A000108.

Riordan array (1-y, y) where y=-(1-sqrt(1+4x^2))/(2x).
Sum_{k=0..n} abs(T(n,k)) = A063886(n). - Philippe Deléham, Oct 06 2006
T(0,0)=1; T(n,k)=0 if k < 0 or if k > n; T(n,0) = -T(n-1,0) - T(n-1,1); T(n,k) = T(n,k-1) - T(n-1,k+1) for k >= 1. - Philippe Deléham, Oct 27 2007
T(2n,0) = A000007(n); T(2n+2,2k+2) = -T(2n+2,2k+1) = (-1)^(n-k)*A039598(n,k); T(2n+1,2k+1) = -T(2n+1,2k) = (-1)^(n-k)*A039599(n,k). - Philippe Deléham, Oct 29 2007
Sum_{k>=0} T(m,k)*T(n,k)*(-1)^k = T(m+n,0) = A105523(m+n). - Philippe Deléham, Jan 24 2010

A131245 A046854^2 as an infinite lower triangular matrix.

Original entry on oeis.org

1, 2, 1, 3, 2, 1, 5, 5, 2, 1, 8, 9, 7, 2, 1, 13, 19, 13, 9, 2, 1, 21, 33, 34, 17, 11, 2, 1, 34, 65, 61, 53, 21, 13, 2, 1, 55, 111, 141, 97, 76, 25, 15, 2, 1, 89, 210, 248, 257, 141, 103, 29, 17, 2, 1, 144, 355, 534, 461, 421, 193, 134, 33, 19, 2, 1

Offset: 0

Gary W. Adamson, Jun 22 2007

Left border = Fibonacci numbers.
Row sums = A131246.
A131243 is the square of the reflection triangle to A046854: A065941.
Row sums of A131243 = (1, 3, 6, 14, 30, 67, 146, 322, 705, 1549, ...).

			First few rows of the triangle:
   1;
   2,  1;
   3,  2,  1;
   5,  5,  2,  1;
   8,  9,  7,  2,  1;
  13, 19, 13,  9,  2,  1;
  21, 33, 34, 17, 11,  2,  1;
  ...

Cf. A131243, A131244, A131246, A046854, A065941.

T(n, k) = binomial((n+k)\2, k);
row(n) = my(m=matrix(n+1, n+1, i, j, T(i-1,j-1))); vector(n+1, i, (m^2)[n+1,i]);
lista(nn) = for (n=0, nn, my(v=row(n)); for (i=1, #v, print1(v[i], ", "));); \\ Michel Marcus, Feb 28 2022

More terms from Michel Marcus, Feb 28 2022

A131270 Triangle T(n,k) = 2*A046854(n,k) - 1, read by rows.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 3, 1, 1, 1, 3, 5, 1, 1, 1, 5, 5, 7, 1, 1, 1, 5, 11, 7, 9, 1, 1, 1, 7, 11, 19, 9, 11, 1, 1, 1, 7, 19, 19, 29, 11, 13, 1, 1, 1, 9, 19, 39, 29, 41, 13, 15, 1, 1, 1, 9, 29, 39, 69, 41, 55, 15, 17, 1, 1, 1, 11, 29, 69, 69, 111, 55, 71, 17, 19, 1, 1

Offset: 0

Gary W. Adamson, Jun 23 2007

Row sums = A131269: {1, 2, 3, 6, 11, 20, 35, 60, 101, 168, ...}.

			First few rows of the triangle:
  1;
  1,  1;
  1,  1,  1;
  1,  3,  1,  1;
  1,  3,  5,  1,  1;
  1,  5,  5,  7,  1,  1;
  1,  5, 11,  7,  9,  1,  1;
  1,  7, 11, 19,  9, 11,  1,  1;
  ...

G. C. Greubel, Rows n = 0..100 of triangle, flattened

Cf. A046854, A065941, A000012, A131268, A131269.

[[2*Binomial(Floor((n+k)/2), k) -1: k in [0..n]]:n in [0..12]]; // G. C. Greubel, Jul 09 2019

Table[2*Binomial[Floor[(n+k)/2], k] - 1, {n,0,12}, {k,0,n}]//Flatten (* G. C. Greubel, Jul 09 2019 *)

T(n,k) = 2*binomial((n+k)\2, k)-1; \\ G. C. Greubel, Jul 09 2019

[[2*binomial(floor((n+k)/2), k) -1 for k in (0..n)] for n in (0..12)] # G. C. Greubel, Jul 09 2019

T(n,k) = 2*A046854(n,k) - 1.

Reversed triangle of A131268.

A131402 2*A007318 - (A046854 + A065941 - A000012).

Original entry on oeis.org

1, 1, 1, 1, 3, 1, 1, 4, 4, 1, 1, 6, 7, 6, 1, 1, 7, 14, 14, 7, 1, 1, 9, 20, 33, 20, 9, 1, 1, 10, 31, 56, 56, 31, 10, 1, 1, 12, 40, 97, 111, 97, 40, 12, 1, 1, 13, 55, 142, 217, 217, 142, 55, 13, 1, 1, 15, 67, 213, 358, 463, 358, 213, 67, 15, 1, 1, 16, 86, 287, 590, 841, 841, 590, 287, 86, 16, 1

Offset: 0

Gary W. Adamson, Jul 07 2007

Row sums = A131403: (1, 2, 5, 10, 21, 44, 93, ...).

			First few rows of the triangle are:
  1;
  1,  1;
  1,  3,  1;
  1,  4,  4,  1;
  1,  6,  7,  6,  1;
  1,  7, 14, 14,  7,  1;
  1,  9, 20, 33, 20,  9,  1;
  1, 10, 31, 56, 56, 31, 10,  1;
  ...

Andrew Howroyd, Table of n, a(n) for n = 0..1274

Row sums are A131403.

Cf. A046854, A065941.

T(n,k) = if(k <= n, 2*binomial(n, k) + 1 - binomial((n + k)\2, k) - binomial(n-(k+1)\2, k\2), 0) \\ Andrew Howroyd, Aug 09 2018

2*A007318 - (A046854 + A065941 - A000012) as infinite lower triangular matrices.

T(n,k) = 2*binomial(n, k) + 1 - binomial(floor((n + k)/2), k) - binomial(n-floor((k+1)/2), floor(k/2)). - Andrew Howroyd, Aug 09 2018

Missing terms inserted and a(55) and beyond from Andrew Howroyd, Aug 09 2018

A153342 Binomial transform of triangle A046854 (shifted).

Original entry on oeis.org

1, 2, 0, 4, 1, 0, 8, 4, 1, 0, 16, 12, 5, 1, 0, 32, 32, 18, 6, 1, 0, 64, 80, 56, 25, 7, 1, 0, 128, 192, 160, 88, 33, 8, 1, 0, 256, 448, 432, 280, 129, 42, 9, 1, 0, 512, 1024, 1120, 832, 450, 180, 52, 10, 1, 0, 1024, 2304, 2816, 2352, 1452, 681, 242, 63, 11, 1, 0

Offset: 0

Gary W. Adamson, Dec 24 2008

Row sums = odd indexed Fibonacci numbers.
Mirror image of triangle in A121462. - Philippe Deléham, Dec 31 2008
Triangle T(n,k), 0 <= k <= n, read by rows given by [2,0,0,0,0,0,0,0,0,0,0,0,...] DELTA [0,1/2,1/2,0,0,0,0,0,0,0,0,0,...] where DELTA is the operator defined in A084938. - Philippe Deléham, Jan 01 2009

			First few rows of the triangle =
    1;
    2,    0;
    4,    1,    0;
    8,    4,    1,   0;
   16,   12,    5,   1,   0;
   32,   32,   18,   6,   1,   0;
   64,   80,   56,  25,   7,   1,  0;
  128,  192,  160,  88,  33,   8,  1,  0;
  256,  448,  432, 280, 129,  42,  9,  1, 0;
  512, 1024, 1120, 832, 450, 180, 52, 10, 1, 0;
  ...

Rigoberto Flórez, Leandro Junes, Luisa M. Montoya, and José L. Ramírez, Counting Subwords in Non-Decreasing Dyck Paths, J. Int. Seq. (2025) Vol. 28, Art. No. 25.1.6. See pp. 8, 19.

Cf. A046854, A001519.

Triangle read by rows, A007318 * A046854 (shifted down 1 row, inserting a "1" at (0,0).
G.f.: (1-y*x)/(1-2*x-y*x+y*x^2). - Philippe Deléham, Mar 27 2012
T(n,k) = 2*T(n-1,l) + T(n-1,k-1) - T(n-2,k-1), T(0,0) = 1, T(1,0) = 2, T(1,1) = 0 and T(n,k) = 0 if k<0 or if k>n. - Philippe Deléham, Mar 27 2012

Second term corrected by Philippe Deléham, Jan 01 2009

A131239 Triangle, T(n,k) = 3A007318(n,k) - 2A046854(n,k), read by rows.

Original entry on oeis.org

1, 1, 1, 1, 4, 1, 1, 5, 7, 1, 1, 8, 12, 10, 1, 1, 9, 24, 22, 13, 1, 1, 12, 33, 52, 35, 16, 1, 1, 13, 51, 85, 95, 51, 19, 1, 1, 16, 64, 148, 180, 156, 70, 22, 1, 1, 17, 88, 212, 348, 336, 238, 92, 25, 1, 1, 20, 105, 320, 560, 714, 574, 344, 117, 28, 1, 1, 21, 135, 425, 920, 1274, 1330, 918, 477, 145, 31, 1

Offset: 0

Gary W. Adamson, Jun 21 2007

Row sums = A074878: (1, 2, 6, 14, 32, 70, 239, ...).

A131238 = 2*A007318 - A046854.

			First few rows of the triangle:
  1;
  1,  1;
  1,  4,  1;
  1,  5,  7,  1;
  1,  8, 12, 10,  1;
  1,  9, 24, 22, 13,  1;
  1, 12, 33, 52, 35, 16, 1;
  ...

G. C. Greubel, Rows n = 0..100 of triangle, flattened

Cf. A007318, A046854, A131238, A074878.

B:=Binomial;; Flat(List([0..12], n-> List([0..n], k-> 3*B(n,k) - 2*B(Int((n+k)/2), k) ))); # G. C. Greubel, Jul 12 2019

B:=Binomial; [3*B(n,k) - 2*B(Floor((n+k)/2), k): k in [0..n], n in [0..12]]; // G. C. Greubel, Jul 12 2019

With[{B=Binomial}, Table[3*B[n,k] - 2*B[Floor[(n+k)/2], k], {n,0,12}, {k,0,n}]]//Flatten (* G. C. Greubel, Jul 12 2019 *)

b=binomial; T(n,k) = 3*b(n,k) - 2*b((n+k)\2, k);
for(n=0,12, for(k=0,n, print1(T(n,k), ", "))) \\ G. C. Greubel, Jul 12 2019

b=binomial; [[3*b(n,k) - 2*b(floor((n+k)/2), k) for k in (0..n)] for n in (0..12)] # G. C. Greubel, Jul 12 2019

T(n,k) = 3*A007318(n,k) - 2*A046854(n,k) as infinite lower triangular matrices.

T(n,k) = 3*binomial(n,k) - 2*binomial(floor((n+k)/2), k). - G. C. Greubel, Jul 12 2019

More terms added by G. C. Greubel, Jul 12 2019

A131240 T(n,k) = 2*A046854(n,k) - I.

Original entry on oeis.org

1, 2, 1, 2, 2, 1, 2, 4, 2, 1, 2, 4, 6, 2, 1, 2, 6, 6, 8, 2, 1, 2, 6, 12, 8, 10, 2, 1, 2, 8, 12, 20, 10, 12, 2, 1, 2, 8, 20, 20, 30, 12, 14, 2, 1, 2, 10, 20, 40, 30, 42, 14, 16, 2, 1, 2, 10, 30, 40, 70, 42, 56, 16, 18, 2, 1, 2, 12, 30, 70, 70, 112, 56, 72, 18, 20, 2, 1

Offset: 0

Gary W. Adamson, Jun 21 2007

Row sums = A001595: (1, 3, 5, 9, 15, 25, 41, 67, ...).

A131241 = 3*A046854 - 2*I.

			First few rows of the triangle:
  1;
  2, 1;
  2, 2,  1;
  2, 4,  2, 1;
  2, 4,  6, 2,  1;
  2, 6,  6, 8,  2, 1;
  2, 6, 12, 8, 10, 2, 1;
  ...

G. C. Greubel, Rows n = 0..100 of triangle, flattened

Cf. A001595, A046854, A131241.

T:= function(n,k)
    if k=n then return 1;
    else return 2*Binomial(Int((n+k)/2), k);
    fi;
  end;
Flat(List([0..12], n-> List([0..n], k-> T(n,k)))); # G. C. Greubel, Jul 12 2019

[k eq n select 1 else 2*Binomial(Floor((n+k)/2), k): k in [0..n], n in [0..12]]; // G. C. Greubel, Jul 12 2019

Table[If[k==n, 1, 2*Binomial[Floor[(n+k)/2], k]], {n, 0, 12}, {k, 0, n}]//Flatten (* G. C. Greubel, Jul 12 2019 *)

T(n,k) = if(k==n, 1, 2*binomial((n+k)\2, k));

def T(n, k):
    if (k==n): return 1
    else: return 2*binomial(floor((n+k)/2), k)
[[T(n, k) for k in (0..n)] for n in (0..12)] # G. C. Greubel, Jul 12 2019

T(n,k) = 2*A046854(n,k) - Identity matrix, where A046854 = Pascal's triangle with repeats by columns.

More terms added by G. C. Greubel, Jul 12 2019

A131375 A007318 + A046854 - A049310.

Original entry on oeis.org

1, 2, 1, 1, 3, 1, 2, 3, 4, 1, 1, 6, 6, 5, 1, 2, 5, 13, 10, 6, 1, 1, 9, 15, 24, 15, 7, 1, 2, 7, 27, 35, 40, 21, 8, 1, 1, 12, 28, 66, 70, 62, 28, 9, 1, 2, 9, 46, 84, 141, 126, 91, 36, 10, 1

Offset: 0

Gary W. Adamson, Jul 04 2007

Row sums = A117591: (1, 3, 5, 10, 19, 37, 72,...).

			First few rows of the triangle are:
1;
2, 1;
1, 3, 1;
2, 3, 4, 1;
1, 6, 6, 5, 1;
2, 5, 13, 10, 6, 1;
1, 9, 15, 24, 15, 7, 1;
...

Cf. A046854, A007318, A049310, A117591.

A007318 + A046854 - A049310 as infinite lower triangular matrices.

A081206 Correlation matrix of triangle A046854.

Original entry on oeis.org

1, 1, 1, 1, 2, 1, 1, 2, 2, 1, 1, 3, 3, 3, 1, 1, 3, 4, 4, 3, 1, 1, 4, 6, 7, 6, 4, 1, 1, 4, 7, 9, 9, 7, 4, 1, 1, 5, 10, 14, 16, 14, 10, 5, 1, 1, 5, 11, 17, 21, 21, 17, 11, 5, 1, 1, 6, 15, 25, 34, 37, 34, 25, 15, 6, 1, 1, 6, 16, 29, 42, 50, 50, 42, 29, 16, 6, 1, 1, 7, 21, 41, 64, 82, 89, 82, 64, 41

Offset: 0

Paul Barry, Mar 11 2003

Main diagonal is A081207.

As a square array, defined by T1(i, j) = Sum_{k=0..min(i, j)} T(i, k)*T(j, k), T(i, j)=A046854(i, j)=binomial(floor((i+j)/2), j).

A124039 Triangle read by rows: T(n, k) = (-1)^floor((n+k+2)/2)(2 - (-1)^(n+k))A046854(n-1, k-1) with T(1, 1) = 3.

Original entry on oeis.org

3, 3, -1, -1, -3, 1, -3, 2, 3, -1, 1, 6, -3, -3, 1, 3, -3, -9, 4, 3, -1, -1, -9, 6, 12, -5, -3, 1, -3, 4, 18, -10, -15, 6, 3, -1, 1, 12, -10, -30, 15, 18, -7, -3, 1, 3, -5, -30, 20, 45, -21, -21, 8, 3, -1, -1, -15, 15, 60, -35, -63, 28, 24, -9, -3, 1

Offset: 1

Roger L. Bagula and Gary W. Adamson, Nov 03 2006

			Triangle begins as:
   3;
   3,  -1;
  -1,  -3,   1;
  -3,   2,   3,  -1;
   1,   6,  -3,  -3,   1;
   3,  -3,  -9,   4,   3,  -1;
  -1,  -9,   6,  12,  -5,  -3,   1;
  -3,   4,  18, -10, -15,   6,   3, -1;
   1,  12, -10, -30,  15,  18,  -7, -3,  1;
   3,  -5, -30,  20,  45, -21, -21,  8,  3, -1;
  -1, -15,  15,  60, -35, -63,  28, 24, -9, -3,  1;

G. C. Greubel, Rows n = 1..50 of the triangle, flattened

Cf. A046854, A066170.

Columns include: (-1)^n*A112030(n-1) (k=1), (-1)^floor((n+1)/2)*A064455(n) (k=2).

A124039:= func< n,k | (-1)^Floor((n+k+2)/2)*(2-(-1)^(n+k))*Binomial(Floor((n+k-2)/2), k-1) + 2*0^(n-1) >;
[A124039(n, k): k in [1..n], n in [1..12]]; // G. C. Greubel, Jan 30 2025

(* First program *)
f[n_, m_, d_]:= If[n==m && n>1 && m>1, 0, If[n==m-1 || n==m+1, -1, If[n==m== 1, 3, 0]]];
M[d_]:= Table[T[n,m,d], {n,d}, {m,d}];
A124039[n_]:= Join[{M[1]}, CoefficientList[Det[M[n] - x*IdentityMatrix[n]], x]];
Table[A124039[n], {n,12}]//Flatten
(* Second program *)
A124039[n_, k_]:= (-1)^Floor[(n+k+2)/2]*(2-(-1)^(n-k))*Binomial[Floor[(n+k- 2)/2], k-1] +2*Boole[n==1];
Table[T[n,k], {n,12}, {k,n}]//Flatten (* G. C. Greubel, Jan 30 2025 *)

@CachedFunction
def t(n,k):
    if n< 0: return 0
    if n==0: return 1 if k == 0 else 0
    h = 3*t(n-1,k) if n==1 else 0
    return t(n-1,k-1) - t(n-2,k) - h
def A124039(n,k): return t(n,k) + 2*0^n
print([[A124039(n,k) for k in range(n+1)] for n in range(13)]) # Peter Luschny, Nov 20 2012

def A124039(n,k): return (-1)^((n+k+2)//2)*(2-(-1)^(n+k))*binomial((n+k-2)//2, k-1) + 2*0^(n-1)
print(flatten([[A124039(n,k) for k in range(1,n+1)] for n in range(1,13)])) # G. C. Greubel, Jan 30 2025

T(n, k) = (-1)^floor((n+k+2)/2)*(2 - (-1)^(n+k))*A046854(n-1,k-1) + 2*[n=1]. - G. C. Greubel, Jan 30 2025

Edited by G. C. Greubel, Jan 30 2025

cp's OEIS Frontend

A106180 Matrix inverse of number triangle A046854.

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A131245 A046854^2 as an infinite lower triangular matrix.

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A131270 Triangle T(n,k) = 2*A046854(n,k) - 1, read by rows.

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A131402 2*A007318 - (A046854 + A065941 - A000012).

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A153342 Binomial transform of triangle A046854 (shifted).

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A131239 Triangle, T(n,k) = 3*A007318(n,k) - 2*A046854(n,k), read by rows.

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A131240 T(n,k) = 2*A046854(n,k) - I.

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A131239 Triangle, T(n,k) = 3A007318(n,k) - 2A046854(n,k), read by rows.

A124039 Triangle read by rows: T(n, k) = (-1)^floor((n+k+2)/2)(2 - (-1)^(n+k))A046854(n-1, k-1) with T(1, 1) = 3.