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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: Jakub Jaroslaw Ciaston

Jakub Jaroslaw Ciaston's wiki page.

Jakub Jaroslaw Ciaston has authored 2 sequences.

A210489 Array read by ascending antidiagonals where row n contains the second partial sums of row n of Pascal's triangle.

Original entry on oeis.org

1, 1, 2, 1, 3, 3, 1, 4, 5, 4, 1, 5, 8, 7, 5, 1, 6, 12, 12, 9, 6, 1, 7, 17, 20, 16, 11, 7, 1, 8, 23, 32, 28, 20, 13, 8, 1, 9, 30, 49, 48, 36, 24, 15, 9, 1, 10, 38, 72, 80, 64, 44, 28, 17, 10, 1, 11, 47, 102, 129, 112, 80, 52, 32, 19, 11, 1, 12, 57, 140, 201, 192, 144, 96, 60, 36, 21, 12
Offset: 0

Views

Author

Jakub Jaroslaw Ciaston, Jan 23 2013

Keywords

Comments

Appears to be a transposed version of A188553 with a leading column of 1's.

Examples

			Table starts:
1,  2,   3,     4,      5,      6,      7,      8,      9,     10
1,  3,   5,     7,      9,     11,     13,     15,     17,     19
1,  4,   8,    12,     16,     20,     24,     28,     32,     36
1,  5,  12,    20,     28,     36,     44,     52,     60,     68
1,  6,  17,    32,     48,     64,     80,     96,    112,    128
1,  7,  23,    49,     80,    112,    144,    176,    208,    240
1,  8,  30,    72,    129,    192,    256,    320,    384,    448
1,  9,  38,   102,    201,    321,    448,    576,    704,    832
1, 10,  47,   140,    303,    522,    769,   1024,   1280,   1536
1, 11,  57,   187,    443,    825,   1291,   1793,   2304,   2816
1, 12,  68,   244,    630,   1268,   2116,   3084,   4097,   5120
1, 13,  80,   312,    874,   1898,   3384,   5200,   7181,   9217
1, 14,  93,   392,   1186,   2772,   5282,   8584,  12381,  16398
1, 15, 107,   485,   1578,   3958,   8054,  13866,  20965,  28779
1, 16, 122,   592,   2063,   5536,  12012,  21920,  34831,  49744
1, 17, 138,   714,   2655,   7599,  17548,  33932,  56751,  84575
1, 18, 155,   852,   3369,  10254,  25147,  51480,  90683, 141326
1, 19, 173,  1007,   4221,  13623,  35401,  76627, 142163, 232009
1, 20, 192,  1180,   5228,  17844,  49024, 112028, 218790, 374172
1, 21, 212,  1372,   6408,  23072,  66868, 161052, 330818, 592962
1, 22, 233,  1584,   7780,  29480,  89940, 227920, 491870, 923780
1, 23, 255,  1817,   9364,  37260, 119420, 317860, 719790,1415650
1, 24, 278,  2072,  11181,  46624, 156680, 437280,1037650,2135440
1, 25, 302,  2350,  13253,  57805, 203304, 593960,1474930,3173090
1, 26, 327,  2652,  15603,  71058, 261109, 797264,2068890,4648020
1, 27, 353,  2979,  18255,  86661, 332167,1058373,2866154,6716910
1, 28, 380,  3332,  21234, 104916, 418828,1390540,3924527,9583064

Links

Crossrefs

Cf. A104734, A132379 (another transposed variant), A188553, A193605.

Programs

  • PARI
    T(n,m) = {sum(k=1, m, k*binomial(n,m-k))}
{ for(n=0, 10, for(m=1, 10, print1(T(n,m), ", ")); print) } \\ Andrew Howroyd, Apr 28 2020

Formula

T(n,k) = A193605(n,k).
T(n,m) = Sum_{k=1..m} k*binomial(n,m-k). - Vladimir Kruchinin, Apr 06 2018

Extensions

Offset corrected and terms a(55) and beyond from Andrew Howroyd, Apr 28 2020

A187801 Pascal's triangle construction method applied to {1,1,2} as an initial term.

Original entry on oeis.org

1, 1, 2, 1, 2, 3, 2, 1, 3, 5, 5, 2, 1, 4, 8, 10, 7, 2, 1, 5, 12, 18, 17, 9, 2, 1, 6, 17, 30, 35, 26, 11, 2, 1, 7, 23, 47, 65, 61, 37, 13, 2, 1, 8, 30, 70, 112, 126, 98, 50, 15, 2, 1, 9, 38, 100, 182, 238, 224, 148, 65, 17, 2, 1, 10, 47, 138, 282, 420, 462, 372
Offset: 2

Views

Author

Jakub Jaroslaw Ciaston, Jan 06 2013

Keywords

Comments

A neighborhood decomposition of triangle graph applied to each node gives three identical sequences (independent of start point) {1,3}.
For star graph (depend of start point) generated sequences are: one time {1,3} and three times {1,1,2}.
Triangle of expansion of (1+x+2*x^2)*(1+x)^n. - Philippe Deléham, Mar 10 2013

Examples

			Triangle begins:
1,1,2;
1,2,3,2;
1,3,5,5,2;
1,4,8,10,7,2;
1,5,12,18,17,9,2;
1,6,17,30,35,26,11,2;
1,7,23,47,65,61,37,13,2;
1,8,30,70,112,126,98,50,15,2;
1,9,38,100,182,238,224,148,65,17,2;
1,10,47,138,282,420,462,372,213,82,19,2;
1,11,57,185,420,702,882,834,585,295,101,21,2;
1,12,68,242,605,1122,1584,1716,1419,880,396,122,23,2;
1,13,80,310,847,1727,2706,3300,3135,2299,1276,518,145,25,2;
From _Philippe Deléham_, Mar 10 2013: (Start)
Row 2: 1+x+2*x^2
Row 3: (1+x+2*x^2)*(1+x) = 1+2*x+3*x^2+2*x^3
Row 4: (1+x+2*x^2)*(1+x)^2 = 1+3*x+5*x^2+5*x^3+2*x^4
Row 5: (1+x+2*x^2)*(1+x)^3 = 1+4*x+8*x^2+10*x^3+7*x^4+2*x^5
(End)

Links

Crossrefs

Cf. A187801, A095660, A107232, A029635, A095660.

Programs

  • Mathematica
    c = {1, 1, 2}; Join[{c}, t = Table[c = Append[c, 0]; c = c + RotateRight[c], {9}]]; Flatten[t] (* T. D. Noe, Mar 11 2013 *)

Formula

For the selection of the initial term: neighborhood decomposition of graph.
For sequence: Pascal's triangle construction method applied to selected initial term.
Row sums: A000079(n+2) = (4, 8, 16, 32, 64, ...). - Philippe Deléham, Mar 10 2013

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