cp's OEIS Frontend

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.

Showing 1-10 of 16 results. Next

A125782 Column 2 of table A125781.

Original entry on oeis.org

1, 3, 8, 23, 76, 294, 1328, 6933, 41350, 278679, 2101433, 17574551, 161740315, 1626733107, 17771416520, 209739328923, 2661301094007, 36148700652162, 523597247829866, 8059284921781891, 131408547139817540
Offset: 0

Views

Author

Paul D. Hanna, Dec 09 2006

Keywords

Crossrefs

Cf. A091351; other columns: A091352, A125783, A125784, A125785, A125786.

Formula

a(n) = A091352(n+1) - 1.

A125783 Column 3 of table A125781; also, equals column 1 of matrix power A091351^2.

Original entry on oeis.org

1, 4, 14, 52, 217, 1033, 5604, 34416, 237328, 1822753, 15473117, 144165763, 1464992791, 16144683412, 191967912402, 2451561765083, 33487399558154, 487448547177703, 7535687673952024, 123349262218035648
Offset: 0

Views

Author

Paul D. Hanna, Dec 09 2006

Keywords

Comments

Column k of triangle A091351 = row sums of matrix power A091351^k for k>=0.

Crossrefs

Cf. A091351; other columns: A091352, A125782, A125784, A125785, A125786.

Formula

a(n) = A091352(n+2) - A091352(n+1) - 1.

A125784 Column 4 of table A125781.

Original entry on oeis.org

1, 5, 21, 91, 433, 2307, 13804, 92433, 688611, 5670713, 51291468, 506502769, 5430460072, 62894124926, 783259655434, 10445143907067, 148592182641759, 2247301621235992, 36021020633412788, 610161098104988668
Offset: 0

Views

Author

Paul D. Hanna, Dec 09 2006

Keywords

Comments

Column k of triangle A091351 = row sums of matrix power A091351^k for k>=0.

Examples

			a(n) = A125783(n) + A125786(n-1) for n>0:
A125783 begins: 1, 4, 14, 52, 217, 1033, 5604, 34416, 237328, ...
and A125786 begins: 1, 7, 39, 216, 1274, 8200, 58017, 451283, ...
term-by-term addition forms this sequence.
This sequence can also be derived from the matrix square A091351^2:
1;
2, [1];
4, [4, 1];
9, [14, 6, 1];
24, [52, 30, 8, 1];
77, [217, 153, 52, 10, 1];
295, [1033, 845, 336, 80, 12, 1];
1329, [5604, 5152, 2294, 625, 114, 14, 1]; ...
The terms enclosed in square barackets sum to equal this sequence.

Crossrefs

Cf. A091351; other columns: A091352, A125782, A125783, A125785, A125786.

Formula

a(n) = Sum_{k=0..n} [A091351^2](n+1,k+1) where A091351^2 is the matrix square of A091351.

A125785 Column 5 of table A125781.

Original entry on oeis.org

1, 6, 29, 141, 739, 4276, 27483, 195978, 1544074, 13371684, 126591212, 1303252476, 14517950305, 174196495882, 2241822436160, 30826098464147, 451299846525541, 7012090426122158, 115289977296253757, 2000463474160276658
Offset: 0

Views

Author

Paul D. Hanna, Dec 09 2006

Keywords

Comments

A091352 equals column 1 of both table A125781 and triangle A091351.

Examples

			A091352 begins: [1, 2, 4, 9, 24, 77, 295, 1329, 6934, 41351, ...];
a(5) = A091352(8) - 2*A091352(7) = 6934 - 2*1329 = 4276.

Crossrefs

Cf. A125781; other columns: A091352, A125782, A125783, A125784, A125786.

Formula

a(n) = A091352(n+2) - 2*A091352(n+1). a(n) = A125782(n+1) - A125783(n+1).

A125786 Column 6 of table A125781.

Original entry on oeis.org

1, 7, 39, 216, 1274, 8200, 58017, 451283, 3847960, 35818351, 362337006, 3965467281, 46749441514, 591291743032, 7993582141984, 115104783083605, 1759853074058289, 28485332959460764, 486811835886953020
Offset: 0

Views

Author

Paul D. Hanna, Dec 09 2006

Keywords

Comments

Column k of triangle A091351 = row sums of matrix power A091351^k for k>=0.

Examples

			a(n) = A125784(n+1) - A125783(n+1) for n>0:
A125784 begins: 1, 5, 21, 91, 433, 2307, 13804, 92433, 688611, ...;
A125783 begins: 1, 4, 14, 52, 217, 1033, 5604, 34416, 237328, ...;
term-by-term differences form this sequence.
This sequence can also be derived from the matrix square A091351^2:
1;
2, 1;
4, 4, [1];
9, 14, [6, 1];
24, 52, [30, 8, 1];
77, 217, [153, 52, 10, 1];
295, 1033, [845, 336, 80, 12, 1];
1329, 5604, [5152, 2294, 625, 114, 14, 1]; ...
the terms enclosed in square barackets sum to equal this sequence.

Crossrefs

Cf. A125781; other columns: A091352, A125782, A125783, A125784, A125785.

Formula

a(n) = Sum_{k=0..n} [A091351^2](n+2,k+2) where A091351^2 is the matrix square of A091351.

A125787 Main diagonal of table A125781.

Original entry on oeis.org

1, 2, 8, 52, 433, 4276, 58017, 855463, 13670443, 235745794, 5122022952, 115172651558, 2700441900575, 66188170219650, 1696382503295129, 52131050479431445, 1631282413260019671, 52321326819923379190
Offset: 0

Views

Author

Paul D. Hanna, Dec 09 2006

Keywords

Crossrefs

Cf. A125781, A125788.

A125788 A diagonal of table A125781: a(n) = A125781(n,n+1).

Original entry on oeis.org

1, 3, 14, 91, 739, 8200, 103545, 1454823, 22446667, 434412701, 8858524146, 190620224636, 4325312221672, 103324196180719, 2949714706867857, 86423553562234994, 2610476419197591903, 81455201815035815833
Offset: 0

Views

Author

Paul D. Hanna, Dec 09 2006

Keywords

Crossrefs

Cf. A125781, A125787.

A125789 Antidiagonal sums of table A125781.

Original entry on oeis.org

1, 2, 4, 9, 23, 68, 234, 935, 4313, 22794, 136851, 925272, 6985389, 58417589, 537217862, 5396822876, 58872576894, 693533824034, 8773792493050, 118464612577309, 1694805159152944, 25478007021649650, 398912310757588476
Offset: 0

Views

Author

Paul D. Hanna, Dec 09 2006

Keywords

Crossrefs

Cf. A125781.

A091352 Row sums of triangle A091351, in which the k-th column lists the row sums of the k-th power of A091351 (when considered as a lower triangular matrix).

Original entry on oeis.org

1, 2, 4, 9, 24, 77, 295, 1329, 6934, 41351, 278680, 2101434, 17574552, 161740316, 1626733108, 17771416521, 209739328924, 2661301094008, 36148700652163, 523597247829867, 8059284921781892, 131408547139817541
Offset: 0

Views

Author

Paul D. Hanna, Jan 02 2004

Keywords

Comments

Equals column 1 of table A125781. Equals row sums and column 0 (shifted) of triangle A127420. - Paul D. Hanna, Feb 11 2007

Links

Crossrefs

Cf. A091351, A091353, A091354.
Cf. A125781, A127420.

A136212 Triple factorial array, read by antidiagonals, where row n+1 is generated from row n by first removing terms in row n at positions {[m*(m+5)/6], m >= 0} and then taking partial sums, starting with all 1's in row 0.

Original entry on oeis.org

1, 1, 1, 4, 2, 1, 28, 10, 3, 1, 280, 80, 18, 4, 1, 3640, 880, 162, 28, 5, 1, 58240, 12320, 1944, 280, 39, 6, 1, 1106560, 209440, 29160, 3640, 418, 52, 7, 1, 24344320, 4188800, 524880, 58240, 5714, 600, 66, 8, 1, 608608000, 96342400, 11022480, 1106560, 95064
Offset: 0

Views

Author

Paul D. Hanna, Dec 22 2007

Keywords

Comments

This is the triple factorial variant of Moessner's factorial array described by A125714 and also of the double factorial array A135876. Another very interesting variant is A136217.

Examples

			Square array begins:
(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),10,11,(12),13,14,(15),16,17,(18),19,20,21,..;
(4),(10),(18),28,(39),52,(66),82,(99),118,138,(159),182,206,(231),258,286,..;
(28),(80),(162),280,(418),600,(806),1064,(1350),1696,2074,(2485),2966,3484,..;
(280),(880),(1944),3640,(5714),8680,(12164),16840,(22194),29080,36824,(45474),.;
(3640),(12320),(29160),58240,(95064),151200,(219108),315440,(428652),581680,...;
(58240),(209440),(524880),1106560,(1864456),3082240,...;
where terms in parenthesis are at positions {[m*(m+5)/6], m>=0}
and are removed before taking partial sums to obtain the next row.
To generate the array, start with all 1's in row 0; from then on,
obtain row n+1 from row n by first removing terms in row n at
positions {[m*(m+5)/6], m>=0} and then taking partial sums.
For example, to generate row 2 from row 1:
[(1),(2),(3),4,(5),6,(7),8,(9),10,11,(12),13,14,(15),16,17,(18),...],
remove terms at positions [0,1,2,4,6,8,11,14,17,...] to get:
[4, 6, 8, 10,11, 13,14, 16,17, 19,20,21, 23,24,25, 27,28,29, ...]
then take partial sums to obtain row 2:
[4, 10, 18, 28,39, 52,66, 82,99, 118,138,159, 182,206,231, ...].
Continuing in this way will generate all the rows of this array.

Crossrefs

Cf. columns: A007559, A008544, A032031, A024216, A024395; related tables: A136213, A125714, A135876, A127054, A125781; A136217.

Programs

  • Mathematica
    t[n_, k_] := t[n, k] = Module[{a = 0, m = 0, c = 0, d = 0}, If[n == 0, a = 1, While[d <= k, If[c == Quotient[(m*(m + 5)), 6], m += 1, a += t[n - 1, c]; d += 1]; c += 1]]; a]; Table[t[n - k, k], {n, 0, 9}, {k, 0, n}] // Flatten (* Jean-François Alcover, Mar 06 2013, translated from Pari *)
  • PARI
    {T(n, k)=local(A=0, m=0, c=0, d=0); if(n==0, A=1, until(d>k, if(c==(m*(m+5))\6, m+=1, A+=T(n-1, c); d+=1); c+=1)); A}

Formula

Columns 0, 1 and 2 form the triple factorials A007559, A008544 and A032031, respectively. Column 4 equals A024216; column 6 equals A024395.
Showing 1-10 of 16 results. Next

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