A054142 -id:A054142 - cp's OEIS frontend

A122073 Triangular array from Steinbach matrices plus their squares as characteristic polynomials: M[i,j]=A[i,j]+A[i,j]^2: A[i,j]^2=Min[i,j]; CharacteristicPolynomial[M[i,j]];.

1, 2, -1, 0, -4, 1, 2, -9, 8, -1, -2, -3, 19, -12, 1, -4, -6, 47, -55, 18, -1, 2, 15, 0, -88, 93, -24, 1, 2, 23, -7, -190, 324, -182, 32, -1, 0, -12, -63, 62, 332, -554, 274, -40, 1, 2, -9, -108, 133, 678, -1642, 1346, -450, 50, -1, -2, -11, 55, 276, -463, -1129, 2832, -2128, 630, -60, 1, -4, -30, 71, 543, -1044, -2204, 7761

Offset: 1

Gary W. Adamson and Roger L. Bagula, Oct 16 2006

Based on the idea that the Steinbach matrices form a "golden Field". Matrices are: {{2, 2}, {2, 2}}, {{2, 2, 2}, {2, 3, 2}, {2, 2, 3}}, {{2, 2, 2, 2}, {2, 3, 3, 2}, {2, 3, 3, 3}, {2, 2, 3, 4}}, {{2, 2, 2, 2, 2}, {2, 3, 3, 3, 2}, {2, 3, 4, 3, 3}, {2, 3, 3, 4, 4}, {2, 2, 3, 4, 5}}, {{2, 2, 2, 2, 2, 2}, {2,3, 3, 3, 3, 2}, {2, 3, 4, 4, 3, 3}, {2, 3, 4, 4, 4, 4}, {2, 3, 3, 4, 5, 5}, {2, 2, 3, 4, 5, 6}}

			{1},
{2, -1},
{0, -4, 1},
{2, -9, 8, -1},
{-2, -3, 19, -12, 1},
{-4, -6,47, -55, 18, -1}
{2, 15, 0, -88, 93, -24, 1},
{2, 23, -7, -190, 324, -182, 32, -1},
{0, -12, -63, 62, 332, -554, 274, -40, 1}

P. Steinbach, Golden fields: a case for the heptagon, Math. Mag. 70 (1997), no. 1, 22-31.

Cf. A038223, A038223, A076756, A054142.

An[d_] := Table[Min[n, m] + If[n + m - 1 > d, 0, 1], {n, 1, d}, {m, 1, d}]; Join[{{1}}, Table[CoefficientList[CharacteristicPolynomial[An[d], x], x], {d,1, 20}]]; Flatten[%]

d-th level M(i,j)->An[d]; T(n,m)=CoefficientList[CharacteristicPolynomial[An[d], x], x]

A171824 Triangle T(n,k)= binomial(n + k,n) + binomial(2*n-k,n) read by rows.

Original entry on oeis.org

2, 3, 3, 7, 6, 7, 21, 14, 14, 21, 71, 40, 30, 40, 71, 253, 132, 77, 77, 132, 253, 925, 469, 238, 168, 238, 469, 925, 3433, 1724, 828, 450, 450, 828, 1724, 3433, 12871, 6444, 3048, 1452, 990, 1452, 3048, 6444, 12871, 48621, 24320, 11495, 5225, 2717, 2717, 5225, 11495, 24320, 48621

Offset: 0

Roger L. Bagula, Dec 19 2009

			Triangle begins as:
       2;
       3,     3;
       7,     6,     7;
      21,    14,    14,    21;
      71,    40,    30,    40,   71;
     253,   132,    77,    77,  132,  253;
     925,   469,   238,   168,  238,  469, 925;
    3433,  1724,   828,   450,  450,  828, 1724,  3433;
   12871,  6444,  3048,  1452,  990, 1452, 3048,  6444, 12871;
   48621, 24320, 11495,  5225, 2717, 2717, 5225, 11495, 24320, 48621;
  184757, 92389, 43824, 19734, 9009, 6006, 9009, 19734, 43824, 92389, 184757;

G. C. Greubel, Table of n, a(n) for n = 0..1325

Row sums are A000984(n+1).

Cf. A001700, A007318, A054142, A085478.

T:= func< n,k | Binomial(n+k,n) + Binomial(2*n-k,n) >;
[T(n,k): k in [0..n], n in [0..12]]; // G. C. Greubel, Apr 29 2021

T[n_, k_] = Binomial[n+k, k] + Binomial[2*n-k, n-k];
Table[T[n, k], {n,0,10}, {k,0,n}]//Flatten

def T(n, k): return binomial(n+k,n) + binomial(2*n-k,n)
flatten([[T(n,k) for k in (0..n)] for n in (0..12)]) # G. C. Greubel, Apr 29 2021

T(n,k) = A046899(n,k) + A092392(n,k).

Sum_{k=0..n} T(n,k) = binomial(2*n+2, n+1) = 2*A001700(n) = A000984(n+1). - G. C. Greubel, Apr 29 2021

Formula and row sums reference added by the Assoc. Editors of the OEIS, Feb 24 2010

A175990 Irregular triangle read by rows: t(n,m) = binomial(n-m-1,m+1) for 0 <= m <= floor((n-1)/2).

Original entry on oeis.org

1, 2, 0, 3, 1, 4, 3, 0, 5, 6, 1, 6, 10, 4, 0, 7, 15, 10, 1, 8, 21, 20, 5, 0, 9, 28, 35, 15, 1, 10, 36, 56, 35, 6, 0, 11, 45, 84, 70, 21, 1, 12, 55, 120, 126, 56, 7, 0, 13, 66, 165, 210, 126, 28, 1, 14, 78, 220, 330, 252, 84, 8, 0, 15, 91, 286, 495, 462, 210, 36, 1, 16, 105, 364, 715, 792, 462, 120, 9, 0, 17, 120

Offset: 2

Roger L. Bagula, Dec 06 2010

			Triangle begins:
   1;
   2,  0;
   3,  1;
   4,  3,  0;
   5,  6,  1;
   6, 10,  4,  0;
   7, 15, 10,  1;
   8, 21, 20,  5,  0;
   9, 28, 35, 15,  1;
  10, 36, 56, 35,  6,  0;
  11, 45, 84, 70, 21,  1;

Row sums are A000071.
Essentially the same as A011973 (removing first column). Elimination of each 2nd row yields essentially A054142 or A121314. Interleaving with zeros gives A052553.
Padding with an initial column of 1's and more zeros yields A169803. Signed variants are A115139 and A124033.

Table[Binomial[n-m-1,m+1],{n,2,15},{m,0,Floor[(n-1)/2]}]//Flatten (* Harvey P. Dale, May 08 2023 *)

Definition clarified by Harvey P. Dale, May 08 2023

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A122073 Triangular array from Steinbach matrices plus their squares as characteristic polynomials: M[i,j]=A[i,j]+A[i,j]^2: A[i,j]^2=Min[i,j]; CharacteristicPolynomial[M[i,j]];.

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A171824 Triangle T(n,k)= binomial(n + k,n) + binomial(2*n-k,n) read by rows.

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A175990 Irregular triangle read by rows: t(n,m) = binomial(n-m-1,m+1) for 0 <= m <= floor((n-1)/2).

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