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 1-7 of 7 results.

A136731 Column 1 of square array A136730.

Original entry on oeis.org

1, 2, 5, 23, 175, 1935, 28432, 523290, 11587072, 299942890, 8886126540, 296438370794, 10993731095695, 448604373236731, 19971257117211555, 963142501803505255, 50015707804752012825, 2782336529985704607295
Offset: 0

Views

Author

Paul D. Hanna, Jan 19 2008

Keywords

Crossrefs

Cf. A136730; A101482, A136732.

A136732 Column 2 of square array A136730.

Original entry on oeis.org

1, 3, 9, 43, 324, 3510, 50528, 913377, 19918602, 509040779, 14918466255, 493115508126, 18143982947900, 735340631600946, 32542320101428755, 1561227609244084205, 80700623119099359600, 4470904603875492038790
Offset: 0

Views

Author

Paul D. Hanna, Jan 19 2008

Keywords

Crossrefs

Cf. A136730; A101482, A136731.

A101482 Column 1 of triangular matrix T=A101479, in which row n equals row (n-1) of T^(n-1) followed by '1'.

Original entry on oeis.org

1, 1, 2, 9, 70, 795, 11961, 224504, 5051866, 132523155, 3969912160, 133678842902, 4997280555576, 205320100093953, 9195224163850830, 445775353262707365, 23255990676521697670, 1299028117862237432959, 77348967890083608924045
Offset: 0

Views

Author

Paul D. Hanna, Jan 21 2005

Keywords

Comments

Number of Dyck paths whose ascent lengths are exactly {n, n-1, .. 1}, for example the a(2) = 2 paths are uududd and uuddud. - David Scambler, May 30 2012

Examples

			This sequence can also be generated in the following manner.
Start a table with the all 1's sequence in row 0; from then on, row n+1 can be formed from row n by dropping the initial n terms of row n and taking partial sums of the remaining terms to obtain row n+1.
The following table (A136730) illustrates this method:
  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, ...;
  [2, 5], 9, 14, 20, 27, 35, 44, 54, 65, 77, 90, 104, ...;
  [9, 23, 43], 70, 105, 149, 203, 268, 345, 435, 539, 658, ...;
  [70, 175, 324, 527], 795, 1140, 1575, 2114, 2772, 3565, ...;
  [795, 1935, 3510, 5624, 8396], 11961, 16471, 22096, 29025, ...;
  [11961, 28432, 50528, 79553, 117020, 164672], 224504, ...; ...
In the above table, drop the initial n terms in row n (enclosed in square brackets) and then take partial sums to obtain row n+1 for n>=0;
this sequence then forms the first column of the resultant table.
Note: column k of the above table equals column 1 of matrix power T^(k+1) where T=A101479, for k>=0.

Crossrefs

Cf. A101479, A101481, A101483, A136730.

Programs

  • PARI
    {a(n)=local(A=Mat(1),B);for(m=1,n+1,B=matrix(m,m);for(i=1,m, for(j=1,i, if(j==i,B[i,j]=1,B[i,j]=(A^(i-1))[i-1,j]);));A=B);return(A[n+1,1])}

Formula

Equals column 0 of array A136730.

A136737 Square array, read by antidiagonals, where T(n,k) = T(n,k-1) + T(n-1,k+n+1) for n>0, k>0, such that T(n,0) = T(n-1,n+1) for n>0 with T(0,k)=1 for k>=0.

Original entry on oeis.org

1, 1, 1, 4, 2, 1, 30, 9, 3, 1, 335, 69, 15, 4, 1, 4984, 769, 118, 22, 5, 1, 92652, 11346, 1317, 178, 30, 6, 1, 2065146, 208914, 19311, 1995, 250, 39, 7, 1, 53636520, 4613976, 352636, 29126, 2820, 335, 49, 8, 1, 1589752230, 118840164, 7722840, 528097, 41061
Offset: 0

Views

Author

Paul D. Hanna, Jan 19 2008

Keywords

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,...;
(1,2,3), 4, 5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,...;
(4,9,15,22), 30, 39,49,60,72,85,99,114,130,147,165,184,204,225,247,...;
(30,69,118,178,250), 335, 434,548,678,825,990,1174,1378,1603,1850,...;
(335,769,1317,1995,2820,3810), 4984, 6362,7965,9815,11935,14349,...;
(4984,11346,19311,29126,41061,55410,72492), 92652, 116262, 143722,...;
(92652,208914,352636,528097,740035,993678,1294776,1649634), 2065146,..;
(2065146,4613976,7722840,11476963,15971180,21310710,27611970,35003430,43626510),..;
where the rows are generated as follows.
Start row 0 with all 1's; from then on,
remove the first n+2 terms (shown in parenthesis) from row n
and then take partial sums to yield row n+1.
Note the second upper diagonal forms column 0 and equals A121413:
[1,1,4,30,335,4984,92652,2065146,53636520,1589752230,52926799310,...].
which equals column 3 of triangle A101479:
1;
1, 1;
1, 1, 1;
3, 2, 1, 1;
19, 9, 3, 1, 1;
191, 70, 18, 4, 1, 1;
2646, 795, 170, 30, 5, 1, 1;
46737, 11961, 2220, 335, 45, 6, 1, 1;
1003150, 224504, 37149, 4984, 581, 63, 7, 1, 1; ...
where row n equals row (n-1) of T^(n-1) with appended '1'.

Crossrefs

Cf. A101479; columns: A121413, A121417, A121422; diagonals: A121427, A136741; variants: A136730, A136733.

Programs

  • PARI
    {T(n,k)=if(k<0,0,if(n==0,1,T(n,k-1) + T(n-1,k+n+1)))}

A136733 Square array, read by antidiagonals, where T(n,k) = T(n,k-1) + T(n-1,k+n) for n>0, k>0, such that T(n,0) = T(n-1,n) for n>0 with T(0,k)=1 for k>=0.

Original entry on oeis.org

1, 1, 1, 3, 2, 1, 18, 7, 3, 1, 170, 43, 12, 4, 1, 2220, 403, 76, 18, 5, 1, 37149, 5188, 711, 118, 25, 6, 1, 758814, 85569, 9054, 1107, 170, 33, 7, 1, 18301950, 1725291, 147471, 13986, 1605, 233, 42, 8, 1, 508907970, 41145705, 2938176, 225363, 20171, 2220, 308
Offset: 0

Views

Author

Paul D. Hanna, Jan 19 2008

Keywords

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,...];
(1,2), 3, 4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,...;
(3,7,12), 18, 25,33,42,52,63,75,88,102,117,133,150,168,187,207,228,...;
(18,43,76,118), 170, 233,308,396,498,615,748,898,1066,1253,1460,...;
(170,403,711,1107,1605), 2220, 2968,3866,4932,6185,7645,9333,11271,...;
(2220,5188,9054,13986,20171,27816), 37149, 48420,61902,77892,96712,...;
(37149,85569,147471,225363,322075,440785,585046), 758814, 966477,...;
(758814,1725291,2938176,4441557,6285390,8526057,11226958,14459138), ...;
where the rows are generated as follows.
Start row 0 with all 1's; from then on,
remove the first n+1 terms (shown in parenthesis) from row n
and then take partial sums to yield row n+1.
Note the first upper diagonal forms column 0 and equals A101483:
[1,1,3,18,170,2220,37149,758814,18301950,508907970,16023271660,...]
which equals column 2 of triangle A101479:
1;
1, 1;
1, 1, 1;
3, 2, 1, 1;
19, 9, 3, 1, 1;
191, 70, 18, 4, 1, 1;
2646, 795, 170, 30, 5, 1, 1;
46737, 11961, 2220, 335, 45, 6, 1, 1;
1003150, 224504, 37149, 4984, 581, 63, 7, 1, 1; ...
where row n equals row (n-1) of T^(n-1) with appended '1'.

Crossrefs

Cf. A101479; columns: A101483, A121418, A121421; A121425 (main diagonal); variants: A136730, A136737.

Programs

  • PARI
    {T(n,k)=if(k<0,0,if(n==0,1,T(n,k-1) + T(n-1,k+n)))}

A152405 Square array, read by antidiagonals, where row n+1 is generated from row n by first removing terms in row n at positions {m*(m+1)/2, m>=0} and then taking partial sums, starting with all 1's in row 0.

Original entry on oeis.org

1, 1, 1, 3, 2, 1, 14, 8, 3, 1, 86, 45, 14, 4, 1, 645, 318, 86, 22, 5, 1, 5662, 2671, 645, 152, 31, 6, 1, 56632, 25805, 5662, 1251, 232, 41, 7, 1, 633545, 280609, 56632, 11869, 2026, 327, 53, 8, 1, 7820115, 3381993, 633545, 126987, 20143, 2991, 457, 66, 9, 1
Offset: 0

Views

Author

Paul D. Hanna, Dec 05 2008

Keywords

Examples

			Table begins:
(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),...;
(3),(8),14,(22),31,41,(53),66,80,95,(112),130,149,169,190,...;
(14),(45),86,(152),232,327,(457),606,775,965,(1202),1464,1752,2067,...;
(86),(318),645,(1251),2026,2991,(4455),6207,8274,10684,(13934),17653,...;
(645),(2671),5662,(11869),20143,30827,(48480),70355,96990,128959,...;
(5662),(25805),56632,(126987),223977,352936,(582183),874664,1240239,...;
(56632),(280609),633545,(1508209),2748448,4438122,(7641111),11831184,...;
(633545),(3381993),7820115,(19651299),36837937,60743909,...; ...
where row n equals the partial sums of row n-1 after removing terms
at positions {m*(m+1)/2, m>=0} (marked by parenthesis in above table).
For example, to generate row 3 from row 2:
[3,8, 14, 22, 31,41, 53, 66,80,95, 112, 130,...]
remove terms at positions {0,1,3,6,10,...}, yielding:
[14, 31,41, 66,80,95, 130,149,169,190, ...]
then take partial sums to obtain row 3:
[14, 45,86, 152,232,327, 457,606,775,965, ...].
Continuing in this way generates all rows of this table.
RELATION TO POWERS OF A SPECIAL TRIANGULAR MATRIX.
Columns 0 and 1 are found in triangle T=A152400, which begins:
1;
1, 1;
3, 2, 1;
14, 8, 3, 1;
86, 45, 15, 4, 1;
645, 318, 99, 24, 5, 1;
5662, 2671, 794, 182, 35, 6, 1;
56632, 25805, 7414, 1636, 300, 48, 7, 1; ...
where column k of T = column 0 of matrix power T^(k+1) for k>=0.
Furthermore, matrix powers of triangle T=A152400 satisfy:
column k of T^(j+1) = column j of T^(k+1) for all j>=0, k>=0.
Column 3 of this square array = column 1 of T^2:
1;
2, 1;
8, 4, 1;
45, 22, 6, 1;
318, 152, 42, 8, 1;
2671, 1251, 345, 68, 10, 1;
25805, 11869, 3253, 648, 100, 12, 1; ...
RELATED TRIANGLE A127714 begins:
1;
1, 1, 1;
1, 2, 2, 3, 3, 3;
1, 3, 5, 5, 8, 11, 11, 14, 14, 14;
1, 4, 9, 14, 14, 22, 33, 44, 44, 58, 72, 72, 86, 86, 86;...
where right border = column 0 of this square array.

Crossrefs

Cf. columns: A127715, A152401, A152404.
Cf. related triangles: A152400, A127714.
Cf. variants: A125781, A127054, A136212, A136217, A135876, A135878, A136730.

Programs

  • 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+1)/2, m+=1, A+=T(n-1, c); d+=1); c+=1)); A}

A165999 Triangle read by rows: T(0,0) = 1, T(n,k) = T(n-1,k-1) + T(n-1,k) for n > 0, 0 < k <= trinv(n), where trinv(n) = floor((1+sqrt(1+8*n))/2), and entries outside triangle are 0.

Original entry on oeis.org

1, 1, 1, 1, 2, 1, 3, 2, 1, 4, 5, 1, 5, 9, 1, 6, 14, 9, 1, 7, 20, 23, 1, 8, 27, 43, 1, 9, 35, 70, 1, 10, 44, 105, 70, 1, 11, 54, 149, 175, 1, 12, 65, 203, 324, 1, 13, 77, 268, 527, 1, 14, 90, 345, 795, 1, 15, 104, 435, 1140, 795, 1, 16, 119, 539, 1575, 1935, 1, 17, 135, 658, 2114
Offset: 0

Views

Author

Gerald McGarvey, Oct 03 2009

Keywords

Comments

There are trinv(n) terms in row n (see A002024). Related to A136730.

Examples

			Triangle begins: [1] [1, 1] [1, 2] [1, 3, 2] [1, 4, 5] [1, 5, 9] [1, 6, 14, 9] [1, 7, 20, 23] [1, 8, 27, 43] [1, 9, 35, 70] [1, 10, 44, 105, 70] [1, 11, 54, 149, 175] [1, 12, 65, 203, 324] [1, 13, 77, 268, 527] [1, 14, 90, 345, 795] [1, 15, 104, 435, 1140, 795]

Crossrefs

A101482 (diagonal T(A000217(n), n))

Programs

  • PARI
    trinv(n) = floor((1+sqrt(1+8*n))/2); f(n) = trinv(n-1); s=19;M=matrix(s,s);for(n=1,s,M[n,1]=1); for(n=2,s,for(k=2,f(n),M[n,k]=M[n-1,k-1]+M[n-1,k])); for(n=1,s,for(k=1,f(n),print1(M[n,k],", ")))
Showing 1-7 of 7 results.

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

Content is available under The OEIS End-User License Agreement.