A134465 -id:A134465 - cp's OEIS frontend

A316101 Sequence a_k of column k shifts left when Weigh transform is applied k times with a_k(n) = n for n<2; square array A(n,k), n>=0, k>=0, read by antidiagonals.

0, 0, 1, 0, 1, 1, 0, 1, 1, 1, 0, 1, 1, 1, 1, 0, 1, 1, 1, 2, 1, 0, 1, 1, 1, 3, 3, 1, 0, 1, 1, 1, 4, 6, 6, 1, 0, 1, 1, 1, 5, 10, 16, 12, 1, 0, 1, 1, 1, 6, 15, 32, 43, 25, 1, 0, 1, 1, 1, 7, 21, 55, 105, 120, 52, 1, 0, 1, 1, 1, 8, 28, 86, 210, 356, 339, 113, 1, 0, 1, 1, 1, 9, 36, 126, 371, 826, 1227, 985, 247, 1

Offset: 0

Alois P. Heinz, Jun 24 2018

			Square array A(n,k) begins:
  0,  0,   0,   0,   0,    0,    0,    0,    0, ...
  1,  1,   1,   1,   1,    1,    1,    1,    1, ...
  1,  1,   1,   1,   1,    1,    1,    1,    1, ...
  1,  1,   1,   1,   1,    1,    1,    1,    1, ...
  1,  2,   3,   4,   5,    6,    7,    8,    9, ...
  1,  3,   6,  10,  15,   21,   28,   36,   45, ...
  1,  6,  16,  32,  55,   86,  126,  176,  237, ...
  1, 12,  43, 105, 210,  371,  602,  918, 1335, ...
  1, 25, 120, 356, 826, 1647, 2961, 4936, 7767, ...

Alois P. Heinz, Antidiagonals n = 0..140, flattened
M. Bernstein and N. J. A. Sloane, Some canonical sequences of integers, Linear Alg. Applications, 226-228 (1995), 57-72; erratum 320 (2000), 210. [Link to arXiv version]
M. Bernstein and N. J. A. Sloane, Some canonical sequences of integers, Linear Alg. Applications, 226-228 (1995), 57-72; erratum 320 (2000), 210. [Link to Lin. Alg. Applic. version together with omitted figures]

Columns k=0-10 give: A057427, A004111, A007561, A316103, A316104, A316105, A316106, A316107, A316108, A316109, A316110.
Rows include (offsets may differ): A000004, A000012, A000027, A000217, A134465.
Main diagonal gives A316102.
Cf. A144042, A316074.

wtr:= proc(p) local b; b:= proc(n, i) option remember;
       `if`(n=0, 1, `if`(i<1, 0, add(binomial(p(i), j)*
         b(n-i*j, i-1), j=0..n/i))) end: j-> b(j$2)
      end:
g:= proc(k) option remember; local b, t; b[0]:= j->
     `if`(j<2, j, b[k](j-1)); for t to k do
       b[t]:= wtr(b[t-1]) od: eval(b[0])
    end:
A:= (n, k)-> g(k)(n):
seq(seq(A(n, d-n), n=0..d), d=0..14);

wtr[p_] := Module[{b}, b[n_, i_] := b[n, i] = If[n == 0, 1, If[i < 1, 0, Sum[Binomial[p[i], j]*b[n - i*j, i - 1], {j, 0, n/i}]]]; b[#, #]&];
g[k_] := g[k] = Module[{b, t}, b[0][j_] := If[j < 2, j, b[k][j - 1]]; For[ t = 1, t <= k + 1, t++, b[t] = wtr[b[t - 1]]]; b[0]];
A[n_, k_] := g[k][n];
Table[A[n, d-n], {d, 0, 14}, {n, 0, d}] // Flatten (* Jean-François Alcover, Jul 10 2018, after Alois P. Heinz *)

A134464 (A127648 * A000012 + A000012 * A127773) - A000012.

Original entry on oeis.org

1, 2, 4, 3, 5, 8, 4, 6, 9, 13, 5, 7, 10, 14, 19, 6, 8, 11, 15, 20, 26, 7, 9, 12, 16, 21, 27, 34, 8, 10, 13, 17, 22, 28, 35, 43, 9, 11, 14, 18, 23, 29, 36, 44, 53, 10, 12, 15, 19, 24, 30, 37, 45, 54, 64

Offset: 1

Gary W. Adamson, Oct 26 2007

Row sums = A134465: (1, 6, 16, 32, 55, 86, ...).

			First few rows of the triangle:
  1;
  2,  4;
  3,  5,  8;
  4,  6,  9, 13;
  5,  7, 10, 14, 19;
  6,  8, 11, 15, 20, 26;
  7,  9, 12, 16, 21, 27, 34;
  ...

Harvey P. Dale, Table of n, a(n) for n = 1..1000

Cf. A127648, A127773, A134465.

Flatten[Table[RecurrenceTable[{a[1]==i,a[n]==a[n-1]+n},a,{n,i}],{i,10}]] (* Harvey P. Dale, Nov 12 2013 *)

(A127648 * A000012 * A000012 * A127773) - A000012, as infinite lower triangular matrices.

A135855 A007318 * a tridiagonal matrix with (1, 4, 1, 0, 0, 0, ...) in every column.

Original entry on oeis.org

1, 5, 1, 10, 6, 1, 16, 16, 7, 1, 23, 32, 23, 8, 1, 31, 55, 55, 31, 9, 1, 40, 86, 110, 86, 40, 10, 1, 50, 126, 196, 196, 126, 50, 11, 1, 61, 176, 322, 392, 322, 176, 61, 12, 1, 73, 237, 498, 714, 714, 498, 237, 73, 13, 1

Offset: 0

Gary W. Adamson, Dec 01 2007

			First few rows of the triangle:
   1;
   5,  1;
  10,  6,   1;
  16, 16,   7,  1;
  23, 32,  23,  8,  1;
  31, 55,  55, 31,  9,  1;
  40, 86, 110, 86, 40, 10, 1;
  ...

G. C. Greubel, Rows n = 0..50 of the triangle, flattened

Cf. A007318, A022815, A052905, A101945, A134465.

A135855:= func< n,k | Binomial(n,k)*(n^2 + (2*k+7)*n - 2*(k^2 + 2*k -1))/((k+1)*(k+2)) >;
[A135855(n,k): k in [0..n], n in [0..10]]; // G. C. Greubel, Feb 06 2022

T[n_, k_]:= T[n, k]= If[k<0 || k>n, 0, If[k==0, (n^2+7*n+2)/2, If[k==n, 1, T[n-1, k-1] + T[n-1, k]]]];
Table[T[n, k], {n,0,12}, {k,0,n}]//Flatten (* G. C. Greubel, Feb 06 2022 *)

@CachedFunction
def T(n,k): # A135855
    if (k==0): return (n^2+7*n+2)/2
    elif (k==n): return 1
    else: return T(n-1, k-1) + T(n-1, k)
flatten([[T(n,k) for k in (0..n)] for n in (0..10)]) # G. C. Greubel, Feb 06 2022

Binomial transform of an infinite tridiagonal matrix with (1, 4, 1, 0, 0, 0, ...) in every column; i.e., (1, 1, 1, ...) in the main diagonal, (4, 4, 4, 0, 0, 0, ...) in the subdiagonal and (1, 1, 1, ...) in the subsubdiagonal.
T(n, 0) = A052905(n).
Sum_{k=0..n} T(n, k) = A101945(n).
From G. C. Greubel, Feb 06 2022: (Start)
T(n, k) = T(n-1, k-1) + T(n-1, k), with T(n, n) = 1, T(n, 0) = A052905(n).
T(n, k) = binomial(n,k)*(n^2 + (2*k+7)*n - 2*(k^2 + 2*k -1))/((k+1)*(k+2)).
T(n, 1) = A134465(n).
T(n, 2) = A022815(n-1).
T(n, n-1) = n+3.
T(n, n-2) = A052905(n+2). (End)

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A316101 Sequence a_k of column k shifts left when Weigh transform is applied k times with a_k(n) = n for n<2; square array A(n,k), n>=0, k>=0, read by antidiagonals.

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A134464 (A127648 * A000012 + A000012 * A127773) - A000012.

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A135855 A007318 * a tridiagonal matrix with (1, 4, 1, 0, 0, 0, ...) in every column.

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