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This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

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A125101 T(n,k) = k*binomial(n-1,k-1) + Fibonacci(k)*binomial(n-1,k) (1 <= k <= n).

Original entry on oeis.org

1, 2, 2, 3, 5, 3, 4, 9, 11, 4, 5, 14, 26, 19, 5, 6, 20, 50, 55, 30, 6, 7, 27, 85, 125, 105, 44, 7, 8, 35, 133, 245, 280, 182, 62, 8, 9, 44, 196, 434, 630, 560, 300, 85, 9, 10, 54, 276, 714, 1260, 1428, 1056, 477, 115, 10, 11, 65, 375, 1110, 2310, 3192, 3030, 1905, 745, 155
Offset: 1

Views

Author

Gary W. Adamson, Nov 20 2006

Keywords

Comments

Row sums are s(n) = 1, 4, 11, 28, 69, 167, 400, ...
Binomial transform of the bidiagonal matrix with (1,2,3...) in the main diagonal and the Fibonacci numbers (1,1,2,3,5,8,...) in the subdiagonal.

Examples

			First few rows of the triangle:
  1;
  2,  2;
  3,  5,   3;
  4,  9,  11,   4;
  5, 14,  26,  19,   5;
  6, 20,  50,  55,  30,   6;
  7, 27,  85, 125, 105,  44,  7;
  8, 35, 133, 245, 280, 182, 62, 8;
  ...

Crossrefs

Cf. A000096, A051925.

Programs

  • Maple
    with(combinat): T:=(n,k)->k*binomial(n-1,k-1)+fibonacci(k)*binomial(n-1,k): for n from 1 to 12 do seq(T(n,k),k=1..n) od; # yields sequence in triangular form
  • Mathematica
    Flatten[Table[k Binomial[n-1,k-1]+Fibonacci[k]Binomial[n-1,k],{n,15},{k,n}]] (* Harvey P. Dale, Nov 03 2014 *)

Formula

T(n,2) = A000096(n-1).
T(n,3) = A051925(n-1).
T(n,4) = A215862(n-3). - R. J. Mathar, Aug 10 2013
Row sums s(n) = 7*s(n-1) -17*s(n-2) +16*s(n-3) -4*s(n-4) with s(n) = A001787(n+1)/4 +A001906(n-1). - R. J. Mathar, Aug 10 2013

Extensions

Edited by N. J. A. Sloane, Nov 29 2006

A264751 Triangle read by rows: T(n,k) is the number of sequences of k <= n throws of an n-sided die (with faces numbered 1, 2, ..., n) in which the sum of the throws first reaches or exceeds n on the k-th throw.

Original entry on oeis.org

1, 1, 2, 1, 5, 3, 1, 9, 11, 4, 1, 14, 26, 19, 5, 1, 20, 50, 55, 29, 6, 1, 27, 85, 125, 99, 41, 7, 1, 35, 133, 245, 259, 161, 55, 8, 1, 44, 196, 434, 574, 476, 244, 71, 9, 1, 54, 276, 714, 1134, 1176, 804, 351, 89, 10, 1, 65, 375, 1110, 2058, 2562, 2190, 1275, 485, 109, 11
Offset: 1

Views

Author

Louis Rogliano, Nov 26 2015

Keywords

Comments

By empirical observation: Sum of rows is A002064.

Examples

			Triangle begins:
  1
  1    2
  1    5    3
  1    9   11    4
  1   14   26   19    5
  1   20   50   55   29    6
  1   27   85  125   99   41    7
  1   35  133  245  259  161   55    8
  1   44  196  434  574  476  244   71    9
  1   54  276  714 1134 1176  804  351   89   10
  1   65  375 1110 2058 2562 2190 1275  485  109   11

Links

Crossrefs

Columns are: A000012 (k=1), A000096 (k=2), A051925 (k=3), A215862 (k=4), A264750 (k=5).
Cf. A007318 (binomial(n-1,k-1) = number of sequences of k throws of an n-sided die in which the sum of the throws equals n).
See also A002064.

Programs

  • Mathematica
    T[n_, k_] := Module[
{i, total = 0, part, perm},
part = IntegerPartitions[n, {k}];
perm = Flatten[Table[Permutations[part[[i]]], {i, 1, Length[part]}],      1];
For[i = 1, i <= Length[perm], i++,    total += n + 1 - perm[[i, k]]    ];
Return[total];   ]
(* The rows are obtained by: *)
g[n_] := Table[T[n,k], {k,1,n}]
(* And the triangle is obtained by: *)
Table[g[n],{n,1,number_of_rows_wanted}]

Formula

Sum_{k = 1..n} T(n,k)*k/n^k = ((n+1)/n)^(n-1) = expected value of k.
Lim_{n->infinity} (expected value of k) = e = 2.71828182845... - Jon E. Schoenfield, Nov 26 2015
T(n,k) = Sum_{i=k..n} i*binomial(i-2,k-2). - Danny Rorabaugh, Mar 04 2016
T(n,n-1) = 2*T(n-1,n-1) + T(n-1,n-2).
By empirical observation, g.f. for column k: (x-k)/(x-1)^(k+1).
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Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

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