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

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

Author

Jakub Jaroslaw Ciaston, Jan 23 2013

Keywords

• nonn
• tabl
• easy

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

• Andrew Howroyd, Table of n, a(n) for n = 0..1325

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