A104989 -id:A104989 - cp's OEIS frontend

A104988 Matrix square of triangle A104980.

1, 2, 1, 8, 4, 1, 42, 20, 6, 1, 266, 120, 38, 8, 1, 1954, 836, 270, 62, 10, 1, 16270, 6616, 2150, 516, 92, 12, 1, 151218, 58576, 19030, 4688, 882, 128, 14, 1, 1551334, 573672, 185674, 46516, 9050, 1392, 170, 16, 1, 17414114, 6159976, 1982310, 502324, 99994, 15956, 2070, 218, 18, 1

Offset: 0

Paul D. Hanna, Apr 10 2005

Triangular matrix A104980 satisfies: SHIFT_LEFT(column 0 of A104980^p) = p*(column p+1 of A104980) for p>=0.

			Triangle begins:
         1;
         2,       1;
         8,       4,       1;
        42,      20,       6,      1;
       266,     120,      38,      8,     1;
      1954,     836,     270,     62,    10,     1;
     16270,    6616,    2150,    516,    92,    12,    1;
    151218,   58576,   19030,   4688,   882,   128,   14,   1;
   1551334,  573672,  185674,  46516,  9050,  1392,  170,  16,  1;
  17414114, 6159976, 1982310, 502324, 99994, 15956, 2070, 218, 18, 1;

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

Cf. A104980, A104982 (column 0), A104989 (row sums).

t[n_, k_]:= t[n, k]= If[k<0 || k>n, 0, If[k==n, 1, If[k==n-1, n, k*t[n, k+1] + Sum[t[j, 0]*t[n, j+k+1], {j, 0, n-k-1}]]]]; (* t = A104980 *)
M:= With[{q=20}, Table[If[j>i, 0, t[i, j]], {i,0,q}, {j,0,q}]];
Table[MatrixPower[M, 2][[n+1, k+1]], {n,0,12}, {k,0,n}]//Flatten (* G. C. Greubel, Jun 07 2021 *)

PARI
```
T(n,k)= if(n
				
```

T(n+1, 0) = 2*A104980(n+3, 3) = 2*A104982(n) for n>=0.

A125777 Moessner triangle based on A000217.

Original entry on oeis.org

1, 3, 6, 13, 28, 21, 69, 161, 137, 55, 433, 1078, 1017, 477, 120, 3141, 8245, 8437, 4460, 1337, 231, 25873, 71008, 77620, 45058, 15415, 3220, 406, 238629, 680451, 786012, 492264, 186729, 44955, 6930, 666, 2436673, 7184170, 8699205, 5804448, 2394150

Offset: 1

Gary W. Adamson, Dec 07 2006

Begin with the triangular numbers A000217 and circle every T(k)-th term, getting the doubly triangular numbers, A002817. Per instructions shown in A125714, take partial sums of the uncircled terms in row 1, denoting this as row 2. Circle the row 2 terms which are one place to the left of row 1 terms. Take partial sums again in analogous operations for subsequent rows.

Left border = A104989: (1, 3, 13, 69, 433...). Right border = the doubly triangular numbers starting (1, 6, 21...): A002817.

			First few rows of the triangle are as follows:
    1;
    3,    6;
   13,   28,   21;
   69,  161,  137,  55;
  433, 1078, 1017, 477, 120;
  ...

J. H. Conway and R. K. Guy, "The Book of Numbers", Springer-Verlag, 1996, p. 64.

Joshua Zucker, Table of n, a(n) for n = 1..55
G. S. Kazandzidis, On a conjecture of Moessner and a general problem, Bull. Soc. Math. Grèce (N.S.) 2 (1961), 23-30.
Dexter Kozen and Alexandra Silva, On Moessner's theorem, Amer. Math. Monthly 120(2) (2013), 131-139.
Calvin T. Long, Strike it out--add it up, Math. Gaz. 66 (438) (1982), 273-277.
Alfred Moessner, Eine Bemerkung über die Potenzen der natürlichen Zahlen, S.-B. Math.-Nat. Kl. Bayer. Akad. Wiss., 29, 1951.
M. Niqui and J. J. M. M. Rutten, A proof of Moessner's theorem by coinduction, High.-Order Symb. Comput. 24(3) (2011), 191-206.
Oskar Perron, Beweis des Moessnerschen Satzes, S.-B. Math.-Nat. Kl. Bayer. Akad. Wiss., 31-34, 1951.

Cf. A125714, A002817, A104989, A000217.

More terms from Joshua Zucker, Jun 17 2007

cp's OEIS Frontend

A104988 Matrix square of triangle A104980.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

PARI

Formula