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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.

User: Tony Foster III

Tony Foster III's wiki page.

Tony Foster III has authored 4 sequences.

A305695 Triangle T(n,k) read by rows: fibonomial coefficients sums triangle.

Original entry on oeis.org

1, 2, 1, 4, 3, 1, 7, 9, 4, 1, 12, 24, 19, 6, 1, 20, 64, 79, 46, 9, 1, 33, 168, 339, 306, 113, 14, 1, 54, 441, 1431, 2126, 1205, 287, 22, 1, 88, 1155, 6072, 14502, 13581, 4928, 736, 35, 1, 143, 3025, 25707, 99587, 149717, 90013, 20371, 1905, 56, 1
Offset: 0

Views

Author

Tony Foster III, Jul 09 2018

Keywords

Comments

The triangle coefficients give sums of Fibonacci powers when multiplied with Lang triangle coefficients and summed (see 2nd formula).

Examples

			n\k|   0    1     2     3      4     5     6    7  8 9
---+--------------------------------------------------
0  |   1
1  |   2    1
2  |   4    3     1
3  |   7    9     4     1
4  |  12   24    19     6      1
5  |  20   64    79    46      9     1
6  |  33  168   339   306    113    14     1
7  |  54  441  1431  2126   1205   287    22    1
8  |  88 1155  6072 14502  13581  4928   736   35  1
9  | 143 3025 25707 99587 149717 90013 20371 1905 56 1

Crossrefs

Cf. A000045, A010048, A000071, A056588, A317360.

Programs

  • PARI
    f(n, k) = prod(j=0, k-1, fibonacci(n-j))/prod(j=1, k, fibonacci(j));
T(n, k) = if (n< 0, 0, T(n-1, k) + f(n+1, k+1));
tabl(nn) = for (n=0, nn, for (k=0, n, print1(T(n, k), ", ")); print); \\ Michel Marcus, Jul 20 2018

Formula

T(n, k) = T(n-1, k) + A010048(n+1, k+1).
Sum_{t=0..n-1} A056588(n-1, n-1-t) * T(k+t, n-1) = Sum_{j=1..k+1} F(j)^n.

Extensions

More terms from Michel Marcus, Jul 20 2018

A317360 Triangle a(n, k) read by rows: coefficient triangle that gives Lucas powers and sums of Lucas powers.

Original entry on oeis.org

1, 1, 2, 1, 7, -4, 1, 24, -23, -8, 1, 76, -164, -79, 16, 1, 235, -960, -1045, 255, 32, 1, 716, -5485, -11155, 5940, 831, -64, 1, 2166, -29816, -116480, 109960, 32778, -2687, -128, 1, 6527, -158252, -1143336, 2024920, 1029844, -176257, -8703, 256, 1, 19628, -822291, -10851888, 34850816, 32711632, -9230829, -937812, 28159, 512
Offset: 0

Views

Author

Tony Foster III, Jul 26 2018

Keywords

Examples

			n\k|  0  1      2       3        4       5         6        7      8     9
---+-------------------------------------------------------------------------
0  |  1
1  |  1  2
2  |  1  7     -4
3  |  1  24    -23     -8
4  |  1  76    -164    -79       16
5  |  1  235   -960    -1045     255     32
6  |  1  716   -5485   -11155    5940    831      -64
7  |  1  2166  -29816  -116480   109960  32778    -2687    -128
8  |  1  6527  -158252 -1143336  2024920 1029844  -176257  -8703   256
9  |  1  19628 -822291 -10851888 4850816 32711632 -9230829 -937812 28159 512

Crossrefs

Cf. A000032, A000045, A055870, A010048, A027961, A005970, A305695.

Programs

  • PARI
    lucas(p)=2*fibonacci(p+1)-fibonacci(p);
S(n, k) = (-1)^floor((k+1)/2)*(prod(j=0, k-1, fibonacci(n-j))/prod(j=1, k, fibonacci(j)));
T(n, k) = sum(j=0, k, lucas(k+1-j)^n * S(n+1, j));
tabl(m) = for (n=0, m, for (k=0, n, print1(T(n, k), ", ")); print);
tabl(9);

Formula

a(n, k) = Sum_{j=0..k} Lucas(k+1-j)^n * A055870(n+1, j).
Sum_{j=0..n} a(n, n-j) * A010048(k-1+j, n) = Lucas(k)^n.
Sum_{j=0..n} a(n, n-j) * A305695(k-2+j, n-1) = Sum_{t=1..k} Lucas(t)^n.

A296229 Triangle read by rows: Eulerian triangle that produces sums of even powers.

Original entry on oeis.org

2, 4, 4, 8, 32, 8, 16, 176, 176, 16, 32, 832, 2112, 832, 32, 64, 3648, 19328, 19328, 3648, 64, 128, 15360, 152448, 309248, 152448, 15360, 128, 256, 63232, 1099008, 3998464, 3998464, 1099008, 63232, 256, 512, 257024, 7479296, 45175808, 79969280, 45175808, 7479296, 257024, 512, 1024, 1037312, 48988160
Offset: 1

Views

Author

Tony Foster III, Feb 14 2018

Keywords

Comments

Finite sums of consecutive even powers are derived from T(n,k) rows and binomial coefficients: Sum_{k=1..n} (2k)^m = Sum_{j=1..m} binomial(n+m+1-j,m+1)*T(m,j).

Examples

			The triangle T(n, k) begins:
n\k |   1     2       3       4       5       6     7   8
----+----------------------------------------------------
  1 |   2
  2 |   4     4
  3 |   8    32       8
  4 |  16   176     176      16
  5 |  32   832    2112     832      32
  6 |  64  3648   19328   19328    3648      64
  7 | 128 15360  152448  309248  152448   15360   128
  8 | 256 63232 1099008 3998464 3998464 1099008 63232 256
...

Crossrefs

Row sums: A000165, A000079, A257609.

Programs

  • Mathematica
    T[n_, k_] := Sum[(-1)^(k-i)*Binomial[n+1, k-i]*(2*i)^(n), {i, 1, k}]; Table[T[n, k], {n, 1, 10}, {k, 1, n}] // Flatten

Formula

T(n,k) = Sum_{i = 1..k} (-1)^(k-i)*binomial(n+1,k-i)*(2*i)^n.
a(n) = 2*A257609(n-1). - Robert G. Wilson v, Feb 19 2018

A280470 Triangle A106534 with reversed rows.

Original entry on oeis.org

1, 1, 2, 2, 3, 5, 5, 7, 10, 15, 14, 19, 26, 36, 51, 42, 56, 75, 101, 137, 188, 132, 174, 230, 305, 406, 543, 731, 429, 561, 735, 965, 1270, 1676, 2219, 2950, 1430, 1859, 2420, 3155, 4120, 5390, 7066, 9285, 12235, 4862, 6292, 8151, 10571, 13726, 17846, 23236, 30302, 39587, 51822, 16796, 21658, 27950, 36101, 46672
Offset: 0

Views

Author

Tony Foster III, Jan 03 2017

Keywords

Examples

			Fibonacci Determinant Triangle:
    1;
    1,    2;
    2,    3,    5;
    5,    7,   10,   15;
   14,   19,   26,   36,   51;
   42,   56,   75,  101,  137,  188;
  132,  174,  230,  305,  406,  543,  731;
  429,  561,  735,  965, 1270, 1676, 2219, 2950;
  ...

Links

Crossrefs

Cf. A000108, A001519, A007317, A011971, A103433, A106534, A197649.

Programs

  • Magma
    &cat [[&+[Binomial(k,j)*Catalan(n-j): j in [0..k]]: k in [0..n]]: n in [0..10]]; // Bruno Berselli, Mar 07 2017
  • Mathematica
    Table[Sum[Binomial[k, j] CatalanNumber[n - j], {j, 0, k}], {n, 0, 10}, {k, 0, n}] // Flatten (* Michael De Vlieger, Mar 08 2017 *)
  • PARI
    C(n)=binomial(2*n,n)/(n+1);
T(n,k)=sum(j=0,k,binomial(k,j)*C(n-j));
for(n=0,10,for(k=0,n,print1(T(n,k),", "));print()); \\ Joerg Arndt, Jan 15 2017

Formula

T(n,k) = Sum_{j=0..k} binomial(k,j) * A000108(n-j). - Joerg Arndt, Jan 15 2017

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