A193605 -id:A193605 - cp's OEIS frontend

A193606 Augmentation of the triangle A193605. See Comments.

1, 1, 3, 1, 7, 17, 1, 12, 57, 127, 1, 18, 134, 531, 1125, 1, 25, 265, 1556, 5513, 11279, 1, 33, 470, 3793, 19152, 62675, 124837, 1, 42, 772, 8175, 55297, 250524, 771121, 1502679, 1, 52, 1197, 16087, 140269, 834879, 3478204, 10185019, 19480445

Offset: 0

Clark Kimberling, Jul 31 2011

For an introduction to the unary operation "augmentation" as applied to triangular arrays or sequences of polynomials, see A193091.

			First five rows of A193606:
1
1...3
1...7....17
1...12...57....127
1...18...134...531...1125

Cf. A193605, A193091.

u[n_, k_] := Sum[Binomial[n, h], {h, 0, k}]
p[n_, k_] := Sum[u[n, h], {h, 0, k}]
Table[p[n, k], {n, 0, 12}, {k, 0, n}] (* A193695 *)
m[n_] := Table[If[i <= j, p[n + 1 - i, j - i], 0], {i, n}, {j, n + 1}]
TableForm[m[4]]
w[0, 0] = 1; w[1, 0] = p[1, 0]; w[1, 1] = p[1, 1];
v[0] = w[0, 0]; v[1] = {w[1, 0], w[1, 1]};
v[n_] := v[n - 1].m[n]
TableForm[Table[v[n], {n, 0, 6}]] (* A193606 *)
Flatten[Table[v[n], {n, 0, 8}]]

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

Jakub Jaroslaw Ciaston, Jan 23 2013

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

			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

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

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

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

T(n,k) = A193605(n,k).

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

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

cp's OEIS Frontend

A193606 Augmentation of the triangle A193605. See Comments.

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

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