A246788 -id:A246788 - cp's OEIS frontend

A246797 Triangle read by rows: T(n,k) is the coefficient A_k in the transformation Sum_{k=0..n} (k+1)x^k = Sum_{k=0..n} A_k(x-2)^k.

1, 5, 2, 17, 14, 3, 49, 62, 27, 4, 129, 222, 147, 44, 5, 321, 702, 627, 284, 65, 6, 769, 2046, 2307, 1404, 485, 90, 7, 1793, 5630, 7683, 5884, 2725, 762, 119, 8, 4097, 14846, 23811, 22012, 12805, 4794, 1127, 152, 9, 9217, 37886, 69891, 75772, 53125, 24954, 7847, 1592, 189, 10

Offset: 0

Derek Orr, Nov 15 2014

Consider the transformation 1 + 2x + 3x^2 + 4x^3 + ... + (n+1)*x^n = A_0*(x-2)^0 + A_1*(x-2)^1 + A_2*(x-2)^2 + ... + A_n*(x-2)^n. This sequence gives A_0, ... A_n as the entries in the n-th row of this triangle, starting at n = 0.

			Triangle starts:
1;
5,        2;
17,      14,     3;
49,      62,    27,     4;
129,    222,   147,    44,     5;
321,    702,   627,   284,    65,     6;
769,   2046,  2307,  1404,   485,    90,    7;
1793,  5630,  7683,  5884,  2725,   762,  119,    8;
4097, 14846, 23811, 22012, 12805,  4794, 1127,  152,   9;
9217, 37886, 69891, 75772, 53125, 24954, 7847, 1592, 189, 10;
...

Cf. A246788, A014106, A000337, A246799.

T(n,k) = (k+1)*sum(i=0,n-k,2^i*binomial(i+k+1,k+1))
for(n=0,10,for(k=0,n,print1(T(n,k),", ")))

T(n,0) = n*2^(n+1)+1, for n >= 0.
T(n,n-1) = n*(2*n+3), for n >= 1.
Row n sums to A014915(n-1) = T(n,0) of A246799.

A246799 Triangle read by rows: T(n,k) is the coefficient A_k in the transformation Sum_{k=0..n} (k+1)x^k = Sum_{k=0..n} A_k(x-3)^k.

Original entry on oeis.org

1, 7, 2, 34, 20, 3, 142, 128, 39, 4, 547, 668, 309, 64, 5, 2005, 3098, 1929, 604, 95, 6, 7108, 13304, 10434, 4384, 1040, 132, 7, 24604, 54128, 51258, 27064, 8600, 1644, 175, 8, 83653, 211592, 234966, 149536, 59630, 15252, 2443, 224, 9, 280483, 802082, 1022286, 761896, 365810, 117312, 25123, 3464, 279, 10

Offset: 0

Derek Orr, Nov 15 2014

Consider the transformation 1 + 2x + 3x^2 + 4x^3 + ... + (n+1)*x^n = A_0*(x-3)^0 + A_1*(x-3)^1 + A_2*(x-3)^2 + ... + A_n*(x-3)^n. This sequence gives A_0, ... A_n as the entries in the n-th row of this triangle, starting at n = 0.

			Triangle starts:
1;
7,           2;
34,         20,       3;
142,       128,      39,      4;
547,       668,     309,     64,      5;
2005,     3098,    1929,    604,     95,      6;
7108,    13304,   10434,   4384,   1040,    132,     7;
24604,   54128,   51258,  27064,   8600,   1644,   175,    8;
83653,  211592,  234966, 149536,  59630,  15252,  2443,  224,   9;
280483, 802082, 1022286, 761896, 365810, 117312, 25123, 3464, 279, 10;
...

Cf. A246797, A014915, A140676, A246788.

T(n, k) = (k+1)*sum(i=0, n-k, 3^i*binomial(i+k+1, k+1))
for(n=0, 10, for(k=0, n, print1(T(n, k), ", ")))

T(n,0) = ((2*n+1)*3^(n+1) + 1)/4, for n >= 0.
T(n,n-1) = n*(3*n+4), for n >= 1.
Row n sums to A014916(n+1) = T(2*n+1,0) of A246788.

A246798 Triangle read by rows: T(n,k) is the coefficient A_k in the transformation Sum_{k=0..n} (k+1)x^k = Sum_{k=0..n} A_k(x+3)^k.

Original entry on oeis.org

1, -5, 2, 22, -16, 3, -86, 92, -33, 4, 319, -448, 237, -56, 5, -1139, 1982, -1383, 484, -85, 6, 3964, -8224, 7122, -3296, 860, -120, 7, -13532, 32600, -33702, 19384, -6700, 1392, -161, 8, 45517, -124864, 150006, -103088, 44330, -12216, 2107, -208, 9, -151313, 465626, -637314, 509272, -261850, 89844, -20573, 3032, -261, 10

Offset: 0

Derek Orr, Nov 15 2014

Consider the transformation 1 + 2x + 3x^2 + 4x^3 + ... + (n+1)*x^n = A_0*(x+3)^0 + A_1*(x+3)^1 + A_2*(x+3)^2 + ... + A_n*(x+3)^n. This sequence gives A_0, ... A_n as the entries in the n-th row of this triangle, starting at n = 0.

			Triangle starts:
1;
-5,           2;
22,         -16,       3;
-86,         92,     -33,       4;
319,       -448,     237,     -56,       5;
-1139,     1982,   -1383,     484,     -85,      6;
3964,     -8224,    7122,   -3296,     860,   -120,      7;
-13532,   32600,  -33702,   19384,   -6700,   1392,   -161,    8;
45517,  -124864,  150006, -103088,   44330, -12216,   2107, -208,    9;
-151313, 465626, -637314,  509272, -261850,  89844, -20573, 3032, -261, 10;
...

Cf. A246788, A045944, A191008.

T(n, k) = (k+1)*sum(i=0, n-k, (-3)^i*binomial(i+k+1, k+1))
for(n=0, 10, for(k=0, n, print1(T(n, k), ", ")))

T(n,0) = (1-(4*n+5)*(-3)^(n+1))/16, for n >= 0.
T(n,n-1) = -n*(3*n+2), for n >= 1.
Row n sums to (-1)^n*A045883(n+1) = T(n,0) of A246788.

cp's OEIS Frontend

A246797 Triangle read by rows: T(n,k) is the coefficient A_k in the transformation Sum_{k=0..n} (k+1)x^k = Sum_{k=0..n} A_k(x-2)^k.

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A246799 Triangle read by rows: T(n,k) is the coefficient A_k in the transformation Sum_{k=0..n} (k+1)x^k = Sum_{k=0..n} A_k(x-3)^k.

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A246798 Triangle read by rows: T(n,k) is the coefficient A_k in the transformation Sum_{k=0..n} (k+1)x^k = Sum_{k=0..n} A_k(x+3)^k.

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cp's OEIS Frontend

A246797 Triangle read by rows: T(n,k) is the coefficient A_k in the transformation Sum_{k=0..n} (k+1)*x^k = Sum_{k=0..n} A_k*(x-2)^k.

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PARI

Formula

A246799 Triangle read by rows: T(n,k) is the coefficient A_k in the transformation Sum_{k=0..n} (k+1)*x^k = Sum_{k=0..n} A_k*(x-3)^k.

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PARI

Formula

A246798 Triangle read by rows: T(n,k) is the coefficient A_k in the transformation Sum_{k=0..n} (k+1)*x^k = Sum_{k=0..n} A_k*(x+3)^k.

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Author

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Programs

PARI

Formula

A246797 Triangle read by rows: T(n,k) is the coefficient A_k in the transformation Sum_{k=0..n} (k+1)x^k = Sum_{k=0..n} A_k(x-2)^k.

A246799 Triangle read by rows: T(n,k) is the coefficient A_k in the transformation Sum_{k=0..n} (k+1)x^k = Sum_{k=0..n} A_k(x-3)^k.

A246798 Triangle read by rows: T(n,k) is the coefficient A_k in the transformation Sum_{k=0..n} (k+1)x^k = Sum_{k=0..n} A_k(x+3)^k.