A103451 -id:A103451 - cp's OEIS frontend

A103452 Inverse of number triangle A103451.

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

Offset: 0

Paul Barry, Feb 06 2005

Triangle T(n,k), 0 <= k <= n, read by rows given by [ -1, 2, 0, 0, 0, 0, 0, ...] DELTA [1, 0, -1/2, 1/2, 0, 0, 0, 0, 0, ...] where DELTA is the operator defined in A084938. - Philippe Deléham, May 01 2007

			Triangle begins
   1;
  -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;
Production matrix begins
  -1, 1;
  -2, 1, 1;
  -2, 1, 0, 1;
  -2, 1, 0, 0, 1;
  -2, 1, 0, 0, 0, 1;
  -2, 1, 0, 0, 0, 0, 1;
  -2, 1, 0, 0, 0, 0, 0, 1;
  -2, 1, 0, 0, 0, 0, 0, 0, 1;

Michael De Vlieger, Table of n, a(n) for n = 0..10010 (Rows 0 <= n <= 140)

Cf. A000007 (row sums), A103453, A103454, A103455.

Table[Range[n] /. {k_ /; k == 1 && n != 1 -> -1, k_ /; k == n -> 1, Integer -> 0}, {n, 15}] // Flatten (* _Michael De Vlieger, Jul 21 2016 *)

flatten([[1 if k==n else -1 if k==0 else 0 for k in (0..n)] for n in (0..15)]) # G. C. Greubel, Jun 18 2021

T(n,k) = 1 if k = n, -1 if k = 0, otherwise 0.
Sum_{k=0..n} T(n, k) = 0^n (row sums).
Sum_{k=0..floor(n/2)} T(n-k, k) = 0^n - (1-(-1)^n)/2 (diagonal sums).
G.f.: (1 - 2*x + y*x^2)/((1-x)*(1-y*x)). - Philippe Deléham, Feb 11 2012

A132047 3A007318 - 2A103451 as infinite lower triangular matrices.

Original entry on oeis.org

1, 1, 1, 1, 6, 1, 1, 9, 9, 1, 1, 12, 18, 12, 1, 1, 15, 30, 30, 15, 1, 1, 18, 45, 60, 45, 18, 1, 1, 21, 63, 105, 105, 63, 21, 1, 1, 24, 84, 168, 210, 168, 84, 24, 1, 1, 27, 108, 252, 378, 378, 252, 108, 27, 1, 1, 30, 135, 360, 630, 756, 630, 360, 135, 30, 1

Offset: 0

Gary W. Adamson, Aug 08 2007

			First few rows of the triangle are:
  1;
  1, 1;
  1, 6, 1;
  1, 9, 9, 1;
  1, 12, 18, 12, 1;
  1, 15, 30, 30, 15, 1;
  1, 18, 45, 60, 45, 18, 1;
  ...

Cf. A007318, A103451, A131128 (row sums).

T(n, k) = my(bnk = binomial(n, k)); 3*bnk - 2*(bnk==1); \\ Michel Marcus, Jun 16 2022

a(n) = 3*A007318(n) - 2*A103451(n).

T(n,k) = 3*C(n,k)-2*(C(n,k-n)+C(n,-k)-C(0,n+k)), 0<=k<=n. [Eric Werley, Jul 01 2011]

Corrected and extended by Roger L. Bagula, Nov 02 2008

A132823 A007318 + 2A103451 - 2A000012.

Original entry on oeis.org

1, 1, 1, 1, 0, 1, 1, 1, 1, 1, 1, 2, 4, 2, 1, 1, 3, 8, 8, 3, 1, 1, 4, 13, 18, 13, 4, 1, 1, 5, 19, 33, 33, 19, 5, 1, 1, 6, 26, 54, 68, 54, 26, 6, 1, 1, 7, 34, 82, 124, 124, 82, 34, 7, 1, 1, 8, 43, 118, 208, 250, 208, 118, 43, 8, 1, 1, 9, 53, 163, 328, 460, 460, 328, 163, 53, 9, 1

Offset: 0

Gary W. Adamson, Sep 02 2007

Row sums = A132824: (1, 2, 2, 4, 10, 24, 54, 116, 242, ...).

			First few rows of the triangle:
  1;
  1, 1;
  1, 0,  1;
  1, 1,  1,  1;
  1, 2,  4,  2,   1;
  1, 3,  8,  8,   3,   1;
  1, 4, 13, 18,  13,   4,  1;
  1, 5, 19, 33,  33,  19,  5,  1;
  1, 6, 26, 54,  68,  54, 26,  6, 1;
  1, 7, 34, 82, 124, 124, 82, 34, 7, 1;
  ...

Cf. A007318, A103451, A000012, A132824.

A(2n,n) gives A115112 for n>0.

A007318 + 2*A103451 - 2*A000012 as infinite lower triangular matrices.

One missing 1 inserted and more terms added by Alois P. Heinz, Feb 10 2019

A135226 Triangle A135225 + A007318 - A103451, read by rows.

Original entry on oeis.org

1, 1, 1, 1, 3, 1, 1, 4, 5, 1, 1, 5, 9, 7, 1, 1, 6, 14, 16, 9, 1, 1, 7, 20, 30, 25, 11, 1, 1, 8, 27, 50, 55, 36, 13, 1, 1, 9, 35, 77, 105, 91, 49, 15, 1, 1, 10, 44, 112, 182, 196, 140, 64, 17, 1, 1, 11, 54, 156, 294, 378, 336, 204, 81, 19, 1

Offset: 0

Gary W. Adamson, Nov 23 2007

Row sums = A083329: (1, 2, 5, 11, 23, 47, 95, ...).

			First few rows of the triangle:
  1;
  1, 1;
  1, 3,  1;
  1, 4,  5,  1;
  1, 5,  9,  7,  1;
  1, 6, 14, 16,  9,  1;
  1, 7, 20, 30, 25, 11,  1;
  1, 8, 27, 50, 55, 36, 13, 1;
...

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

Cf. A007318, A083329, A103451, A135225.

T:= function(n,k)
    if k=0 or k=n then return 1;
    else return ((n+k)/n)*Binomial(n,k);
    fi; end;
Flat(List([0..10], n-> List([0..n], k-> T(n,k) ))); # G. C. Greubel, Nov 20 2019

T:= func< n,k | k eq 0 or k eq n select 1 else ((n+k)/n)*Binomial(n,k) >;
[T(n,k): k in [0..n], n in [0..10]]; // G. C. Greubel, Nov 20 2019

T:= proc(n, k) option remember;
      if k=0 or k=n then 1
    else ((n+k)/n)*binomial(n,k)
      fi; end:
seq(seq(T(n, k), k=0..n), n=0..10); # G. C. Greubel, Nov 20 2019

T[n_, k_]:= T[n, k]= If[k==0 || k==n, 1, ((n+k)/n) Binomial[n, k]]; Table[T[n, k], {n,0,10}, {k,0,n}]//Flatten (* G. C. Greubel, Nov 20 2019 *)

T(n,k) = if(k==0 || k==n, 1, ((n+k)/n)*binomial(n,k)); \\ G. C. Greubel, Nov 20 2019

@CachedFunction
def T(n,k):
    if (k==0 or k==n): return 1
    else: return ((n+k)/n)*binomial(n, k)
[[T(n,k) for k in (0..n)] for n in (0..10)] # G. C. Greubel, Nov 20 2019

T(n,k) = A135225(n,k) + A007318(n,k) - A103451(n,k) as infinite lower triangular matrices.

T(n,k) = ((n+k)/n)*binomial(n,k) with T(n,0) = T(n,n) = 1. - G. C. Greubel, Nov 20 2019

Corrected and extended by Philippe Deléham, Nov 14 2011

A134868 A103451 * A002260.

Original entry on oeis.org

1, 2, 2, 2, 2, 3, 2, 2, 3, 4, 2, 2, 3, 4, 5, 2, 2, 3, 4, 5, 6, 2, 2, 3, 4, 5, 6, 7, 2, 2, 3, 4, 5, 6, 7, 8, 2, 2, 3, 4, 5, 6, 7, 8, 9, 2, 2, 3, 4, 5, 6, 7, 8, 9, 10, 2, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 2, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 2, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13

Offset: 1

Gary W. Adamson, Nov 14 2007

Row sums = A134869: (1, 4, 7, 11, 16, 22, 29, ...).

			First few rows of the triangle:
  1;
  2, 2;
  2, 2, 3;
  2, 2, 3, 4;
  2, 2, 3, 4, 5;
  ...

Michael De Vlieger, Table of n, a(n) for n = 1..10000

Cf. A002260, A071324, A134869.

Table[k + Boole[k == 1 && n != 2], {n, 2, 14}, {k, n - 1}] // Flatten (* Michael De Vlieger, Jul 19 2016 *)

A103451 * A002260 as infinite lower triangular matrices.

Left border of A002260, (1, 1, 1, 1, ...) is replaced by (1, 2, 2, 2, ...).

A135223 Triangle A000012 * A127648 * A103451, read by rows.

Original entry on oeis.org

1, 3, 2, 6, 2, 3, 10, 2, 3, 4, 15, 2, 3, 4, 5, 21, 2, 3, 4, 5, 6, 28, 2, 3, 4, 5, 6, 7, 36, 2, 3, 4, 5, 6, 7, 8, 45, 2, 3, 4, 5, 6, 7, 8, 9, 55, 2, 3, 4, 5, 6, 7, 8, 9, 10, 66, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 78, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 91, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13

Offset: 1

Gary W. Adamson, Nov 23 2007

Row sums = A028387.

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

G. C. Greubel, Rows n = 1..100 of triangle, flattened

Cf. A002260, A103451, A127648.

T:= function(n,k)
    if k=1 then return Binomial(n+1,2);
    else return k;
    fi; end;
Flat(List([1..15], n-> List([1..n], k-> T(n,k) ))); # G. C. Greubel, Nov 20 2019

[k eq 1 select Binomial(n+1,2) else k: k in [1..n], n in [1..15]]; // G. C. Greubel, Nov 20 2019

seq(seq( `if`(k=1, binomial(n+1,2), k), k=1..n), n=1..15); # G. C. Greubel, Nov 20 2019

T[n_, k_]:= T[n, k]= If[k==1, Binomial[n+1, 2], k]; Table[T[n, k], {n, 15}, {k,n}]//Flatten (* G. C. Greubel, Nov 20 2019 *)

T(n,k) = if(k==1, binomial(n+1,2), k); \\ G. C. Greubel, Nov 20 2019

@CachedFunction
def T(n,k):
    if (k==1): return binomial(n+1, 2)
    else: return k
[[T(n,k) for k in (1..n)] for n in (1..15)] # G. C. Greubel, Nov 20 2019

T(n,k) = A000012(n,k) * A127648(n,k) * A103451(n,k) as infinite lower triangular matrices. Replace left border of 1's in A002260 with (1, 3, 6, 10, 15, ...).

T(n, k) = k with T(n,1) = binomial(n+1, 2). - G. C. Greubel, Nov 20 2019

More terms added by G. C. Greubel, Nov 20 2019

A135224 Triangle A103451 * A007318 * A000012, read by rows. T(n, k) for 0 <= k <= n.

Original entry on oeis.org

1, 3, 1, 5, 3, 1, 9, 7, 4, 1, 17, 15, 11, 5, 1, 33, 31, 26, 16, 6, 1, 65, 63, 57, 42, 22, 7, 1, 129, 127, 120, 99, 64, 29, 8, 1, 257, 255, 247, 219, 163, 93, 37, 9, 1, 513, 511, 502, 466, 382, 256, 130, 46, 10, 1

Offset: 0

Gary W. Adamson, Nov 23 2007

Row sums = A132750: (1, 4, 9, 21, 49, 113, ...).

Left border = A083318: (1, 3, 5, 9, 17, 33, ...).

			First few rows of the triangle:
   1;
   3,  1;
   5,  3,  1;
   9,  7,  4,  1;
  17, 15, 11,  5,  1;
  33, 31, 26, 16,  6,  1;
  65, 63, 57, 42, 22,  7,  1;
...

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

Cf. A083318, A103451, A132750.

function T(n,k)
  if k eq 0 and n eq 0 then return 1;
  elif k eq 0 then return 2^n +1;
  else return (&+[Binomial(n, k+j): j in [0..n]]);
  end if; return T; end function;
[T(n,k): k in [0..n], n in [0..10]]; // G. C. Greubel, Nov 20 2019

T:= proc(n, k) option remember;
      if k=0 and n=0 then 1
    elif k=0 then 2^n +1
    else add(binomial(n, k+j), j=0..n)
      fi; end:
seq(seq(T(n, k), k=0..n), n=0..10); # G. C. Greubel, Nov 20 2019

T[n_, k_]:= T[n, k] = If[k==n==0, 1, If[k==0, 2^n +1, Sum[Binomial[n, k + j], {j, 0, n}]]]; Table[T[n, k], {n, 0, 10}, {k, 0, n}]//Flatten (* G. C. Greubel, Nov 20 2019 *)

T(n,k) = if(k==0 && n==0, 1, if(k==0, 2^n +1, sum(j=0, n, binomial(n, k+j)) )); \\ G. C. Greubel, Nov 20 2019

def T(n, k):
    if (k==0 and n==0): return 1
    elif (k==0): return 2^n + 1
    else: return sum(binomial(n, k+j) for j in (0..n))
[[T(n, k) for k in (0..n)] for n in (0..10)] # G. C. Greubel, Nov 20 2019

T(n, k) = A103451(n,k) * A007318(n,k) * A000012(n,k) as infinite lower triangular matrices.
T(n, k) = Sum_{j=0..n} binomial(n, k+j), with T(0,0) = 1 and T(n,0) = 2^n + 1. - G. C. Greubel, Nov 20 2019
T(n, k) = binomial(n, k)*hypergeom([1, k-n], [k+1], -1) - binomial(n, k+n+1)* hypergeom([1, k+1], [k+n+2], -1) + 0^k - 0^n. - Peter Luschny, Nov 20 2019

A135228 Triangle A000012(signed) * A007318 * A103451, read by rows.

Original entry on oeis.org

1, 1, 1, 3, 1, 1, 5, 2, 2, 1, 11, 2, 4, 3, 1, 21, 3, 6, 7, 4, 1, 43, 3, 9, 13, 11, 5, 1, 85, 4, 12, 22, 24, 16, 6, 1, 171, 4, 16, 34, 46, 40, 22, 7, 1, 341, 5, 20, 50, 80, 86, 62, 29, 8, 1, 683, 5, 25, 70, 130, 166, 148, 91, 37, 9, 1, 1365, 6, 30, 95, 200, 296, 314, 239, 128, 46, 10, 1

Offset: 0

Gary W. Adamson, Nov 23 2007

Row sums = A000975: (1, 2, 5, 10, 21, 42, 85, 170, ...).

Left border = A001045: (1, 1, 3, 5, 11, 21, 43, ...).

			First few rows of the triangle are:
    1;
    1, 1;
    3, 1,  1;
    5, 2,  2,  1;
   11, 2,  4,  3,  1;
   21, 3,  6,  7,  4,  1;
   43, 3,  9, 13, 11,  5,  1;
   85, 4, 12, 22, 24, 16,  6, 1;
  171, 4, 16, 34, 46, 40, 22, 7, 1;
...

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

Cf. A000975, A001045, A007318, A103451.

function T(n,k)
  if k eq 0 then return (2^(n+1) +(-1)^n)/3;
  else return (&+[Binomial(n-2*j-1, k-1): j in [0..Floor((n-1)/2)]]);
  end if; return T; end function;
[T(n,k): k in [0..n], n in [0..12]]; // G. C. Greubel, Nov 20 2019

T:= proc(n, k) option remember;
      if k=0 then (2^(n+1) +(-1)^n)/3
    else add(binomial(n-2*j-1, k-1), j=0..floor((n-1)/2))
      fi; end:
seq(seq(T(n, k), k=0..n), n=0..12); # G. C. Greubel, Nov 20 2019

T[n_, k_]:= T[n, k]= If[k==0, (2^(n+1) +(-1)^n)/3, Sum[Binomial[n-1-2*j, k-1], {j,0,Floor[(n-1)/2]}]]; Table[T[n, k], {n,0,12}, {k,0,n}]//Flatten (* G. C. Greubel, Nov 20 2019 *)

T(n,k) = if(k==0, (2^(n+1) +(-1)^n)/3, sum(j=0, (n-1)\2, binomial( n-2*j-1, k-1)) ); \\ G. C. Greubel, Nov 20 2019

@CachedFunction
def T(n, k):
    if (k==0): return (2^(n+1) +(-1)^n)/3
    else: return sum(binomial(n-2*j-1, k-1) for j in (0..floor((n-1)/2)))
[[T(n, k) for k in (0..n)] for n in (0..12)] # G. C. Greubel, Nov 20 2019

T(n,k) = A000012(signed) * A007318 * A103451 as infinite lower triangular matrices. A000012(signed) = (1; -1,1; 1,-1,1; ...).
T(n,k) = Sum_{j=0..floor((n-1)/2)} binomial(n-2*j-1, k-1), with T(n,0) =
(2^(n+1) - (-1)^(n+1))/3 (Jacobsthal_{n+1}).- G. C. Greubel, Nov 20 2019

Offset changed by G. C. Greubel, Nov 20 2019

A135229 Triangle A000012(signed) * A103451 * A007318, read by rows.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 2, 2, 1, 1, 2, 4, 3, 1, 1, 3, 6, 7, 4, 1, 1, 3, 9, 13, 11, 5, 1, 1, 4, 12, 22, 24, 16, 6, 1, 1, 4, 16, 34, 46, 40, 22, 7, 1, 1, 5, 20, 50, 80, 86, 62, 29, 8, 1

Offset: 0

Gary W. Adamson, Nov 23 2007

row sums = A005578 starting (1, 2, 3, 6, 11, 22, 43, 86, ...).

			First few rows of the triangle are:
  1;
  1, 1;
  1, 1,  1;
  1, 2,  2,  1;
  1, 2,  4,  3,  1;
  1, 3,  6,  7,  4,  1;
  1, 3,  9, 13, 11,  5,  1;
  1, 4, 12, 22, 24, 16,  6, 1;
  1, 4, 16, 34, 46, 40, 22, 7, 1;
...

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

Cf. A005578, A007318, A103451.

function T(n,k)
  if k eq 0 then return 1;
  else return (&+[Binomial(n-2*j-1, k-1): j in [0..Floor((n-1)/2)]]);
  end if; return T; end function;
[T(n,k): k in [0..n], n in [0..12]]; // G. C. Greubel, Nov 20 2019

T:= proc(n, k) option remember;
      if k=0 then 1
    else add(binomial(n-2*j-1, k-1), j=0..floor((n-1)/2))
      fi; end:
seq(seq(T(n, k), k=0..n), n=0..12); # G. C. Greubel, Nov 20 2019

T[n_, k_]:= T[n, k]= If[k==0, 1, Sum[Binomial[n-1-2*j, k-1], {j, 0, Floor[(n-1)/2]}]]; Table[T[n, k], {n,0,12}, {k,0,n}]//Flatten (* G. C. Greubel, Nov 20 2019 *)

T(n,k) = if(k==0, 1, sum(j=0, (n-1)\2, binomial( n-2*j-1, k-1)) ); \\ G. C. Greubel, Nov 20 2019

@CachedFunction
def T(n, k):
    if (k==0): 1
    else: return sum(binomial(n-2*j-1, k-1) for j in (0..floor((n-1)/2)))
[[T(n, k) for k in (0..n)] for n in (0..12)] # G. C. Greubel, Nov 20 2019

T(n,k) = A000012(signed) * A103451 * A007318 as infinite lower triangular matrices, where A000012(signed) = (1; -1,1; 1,-1,1; ...).

T(n,k) = Sum_{j=0..floor((n-1)/2)} binomial(n-2*j-1, k-1), with T(n,0) = 1. - G. C. Greubel, Nov 20 2019

Offset changed by G. C. Greubel, Nov 20 2019

A135230 Triangle A103451 * A000012(signed) * A007318, read by rows.

Original entry on oeis.org

1, 1, 1, 2, 1, 1, 1, 2, 2, 1, 2, 2, 4, 3, 1, 1, 3, 6, 7, 4, 1, 2, 3, 9, 13, 11, 5, 1, 1, 4, 12, 22, 24, 16, 6, 1, 2, 4, 16, 34, 46, 40, 22, 7, 1, 1, 5, 20, 50, 80, 86, 62, 29, 8, 1, 2, 5, 25, 70, 130, 166, 148, 91, 37, 9, 1, 1, 6, 30, 95, 200, 296, 314, 239, 128, 46, 10, 1

Offset: 0

Gary W. Adamson, Nov 23 2007

row sums = A135231

			First few rows of the triangle are:
  1;
  1, 1;
  2, 1,  1;
  1, 2,  2,  1;
  2, 2,  4,  3,  1;
  1, 3,  6,  7,  4,  1;
  2, 3,  9, 13, 11,  5,  1;
  1, 4, 12, 22, 24, 16,  6, 1;
  2, 4, 16, 34, 46, 40, 22, 7, 1;
...

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

Cf. A000012, A007318, A103451, A135231.

function T(n,k)
  if k eq n then return 1;
  elif k eq 0 then return (3+(-1)^n)/2;
  else return (&+[Binomial(n-2*j-1, k-1): j in [0..Floor((n-1)/2)]]);
  end if; return T; end function;
[T(n,k): k in [0..n], n in [0..12]]; // G. C. Greubel, Nov 20 2019

T:= proc(n, k) option remember;
      if k=n then 1
    elif k=0 then (3+(-1)^n)/2
    else add(binomial(n-2*j-1, k-1), j=0..floor((n-1)/2))
      fi; end:
seq(seq(T(n, k), k=0..n), n=0..12); # G. C. Greubel, Nov 20 2019

T[n_, k_]:= T[n, k]= If[k==n, 1, If[k==0, (3+(-1)^n)/2, Sum[Binomial[n-1 - 2*j, k-1], {j, 0, Floor[(n-1)/2]}] ]]; Table[T[n, k], {n, 0, 12}, {k, 0, n}]//Flatten (* G. C. Greubel, Nov 20 2019 *)

T(n,k) = if(k==n, 1, if(k==0, (3+(-1)^n)/2, sum(j=0, (n-1)\2, binomial( n-2*j-1, k-1)) )); \\ G. C. Greubel, Nov 20 2019

@CachedFunction
def T(n, k):
    if (k==n): return 1
    elif (k==0): return (3+(-1)^n)/2
    else: return sum(binomial(n-2*j-1, k-1) for j in (0..floor((n-1)/2)))
[[T(n, k) for k in (0..n)] for n in (0..12)] # G. C. Greubel, Nov 20 2019

T(n,k) = A103451 * A000012(signed) * A007318, where A000012(signed) = (1; -1,1; 1,-1,1;...).

T(n,k) = Sum_{j=0..floor((n-1)/2)} binomial(n-2*j-1, k-1), with T(n,0) = (3+(-1)^n)/2 and T(n,n) = 1. - G. C. Greubel, Nov 20 2019

More terms and offset changed by G. C. Greubel, Nov 20 2019

cp's OEIS Frontend

A103452 Inverse of number triangle A103451.

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A132047 3*A007318 - 2*A103451 as infinite lower triangular matrices.

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A132823 A007318 + 2*A103451 - 2*A000012.

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A135226 Triangle A135225 + A007318 - A103451, read by rows.

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A134868 A103451 * A002260.

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A135223 Triangle A000012 * A127648 * A103451, read by rows.

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A135224 Triangle A103451 * A007318 * A000012, read by rows. T(n, k) for 0 <= k <= n.

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A132047 3A007318 - 2A103451 as infinite lower triangular matrices.

A132823 A007318 + 2A103451 - 2A000012.