A248826 -id:A248826 - cp's OEIS frontend

A242598 Triangle read by rows: T(n,k) is the coefficient A_k in the transformation of 1 + x + x^2 + ... + x^n to the polynomial A_k*(x-k)^k for 0 <= k <= n.

1, 2, 1, 2, 5, 1, 2, 14, 10, 1, 2, 30, 58, 17, 1, 2, 55, 258, 167, 26, 1, 2, 91, 978, 1247, 386, 37, 1, 2, 140, 3330, 7862, 4306, 772, 50, 1, 2, 204, 10498, 44150, 40146, 11972, 1394, 65, 1, 2, 285, 31234, 227858, 330450, 153722, 28610, 2333, 82, 1, 2, 385, 88834, 1102658, 2480850, 1728722, 482210, 61133, 3682, 101, 1

Offset: 0

Derek Orr, Oct 15 2014

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

			1;
2,  1;
2,  5,      1;
2,  14,    10,       1;
2,  30,    58,      17,       1;
2,  55,   258,     167,      26,       1;
2,  91,   978,    1247,     386,      37,      1;
2, 140,  3330,    7862,    4306,     772,     50,     1;
2, 204, 10498,   44150,   40146,   11972,   1394,    65,    1;
2, 285, 31234,  227858,  330450,  153722,  28610,  2333,   82,   1;
2, 385, 88834, 1102658, 2480850, 1728722, 482210, 61133, 3682, 101, 1

Cf. A000330, A002522, A138911, A248826.

for(n=0,20,for(k=0,n,if(!k,if(n,print1(2,", "));if(!n,print1(1,", ")));if(k,print1(sum(i=1,n,(k^(i-k)*i*binomial(i,k)))/k,", "))))

T(n,1) = n*(2*n+1)*(n+1)/6 for n > 0.
T(n,n-1) = n^2 + 1 for n > 0.
Rows sum to SUM{k=0..n} A138911(k).

A248829 Triangle read by rows: T(n,k) is the coefficient A_k in the transformation of 1 + x + x^2 + ... + x^n to the polynomial A_k*(x+2k)^k for 0 <= k <= n .

Original entry on oeis.org

1, -1, 1, -1, -7, 1, -1, 29, -17, 1, -1, -99, 175, -31, 1, -1, 301, -1425, 569, -49, 1, -1, -851, 10095, -8071, 1391, -71, 1, -1, 2285, -65169, 97769, -29969, 2869, -97, 1, -1, -5907, 393583, -1063447, 543471, -86731, 5279, -127, 1, -1, 14829, -2260625, 10693865, -8746257, 2181269, -212449, 8945, -161, 1, -1, -36371, 12484975, -101280535, 128879343, -48218731, 7045151, -461455, 14239, -199, 1

Offset: 0

Derek Orr, Oct 15 2014

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

			1;
-1,     1;
-1,    -7,        1;
-1,    29,      -17,        1;
-1,   -99,      175,      -31,        1;
-1,   301,    -1425,      569,      -49,       1;
-1,  -851,    10095,    -8071,     1391,     -71,       1;
-1,  2285,   -65169,    97769,   -29969,    2869,     -97,    1;
-1, -5907,   393583, -1063447,   543471,  -86731,    5279, -127,    1;
-1, 14829, -2260625, 10693865, -8746257, 2181269, -212449, 8945, -161, 1;

Cf. A056220, A248826, A248830.

for(n=0,20,for(k=0,n,if(!k,if(n,print1(-1,", "));if(!n,print1(1,", ")));if(k,print1(sum(i=1,n,((-2*k)^(i-k)*i*binomial(i,k)))/k,", "))))

T(n,n-1) = 1 - 2*n^2 for n > 0.

A248977 Triangle read by rows: T(n,k) is the coefficient A_k in the transformation of 1 + x + x^2 + ... + x^n to the polynomial A_k*(x+3k)^k for 0 <= k <= n.

Original entry on oeis.org

1, -2, 1, -2, -11, 1, -2, 70, -26, 1, -2, -362, 406, -47, 1, -2, 1663, -4994, 1303, -74, 1, -2, -7085, 53326, -27857, 3166, -107, 1, -2, 28636, -518210, 507958, -102674, 6508, -146, 1, -2, -111332, 4707262, -8310026, 2800366, -295892, 11950, -191, 1, -2, 420109, -40642370, 125613106, -67743506, 11185858, -722882, 20221, -242, 1

Offset: 0

Derek Orr, Oct 18 2014

Consider the transformation 1 + x + x^2 + x^3 + ... + x^n = A_0*(x+0)^0 + A_1*(x+3)^1 + A_2*(x+6)^2 + ... + A_n*(x+3n)^n. This sequence gives A_0, ... A_n as the entries in the n-th row of this triangle, starting at n = 0.

			1;
-2,       1;
-2,     -11,       1;
-2,      70,     -26,        1;
-2,    -362,     406,      -47,       1;
-2,    1663,   -4994,     1303,     -74,       1;
-2,   -7085,   53326,   -27857,    3166,    -107,     1;
-2,   28636, -518210,   507958, -102674,    6508,  -146,    1;
-2, -111332, 4707262, -8310026, 2800366, -295892, 11950, -191, 1;

Cf. A248811, A248829, A248826.

for(n=0, 10, for(k=0, n, if(!k, if(n, print1(-2, ", ")); if(!n, print1(1, ", "))); if(k, print1(sum(i=1, n, ((-3*k)^(i-k)*i*binomial(i,k)))/k, ", "))))

T(n,n-1) = 1 - 3n^2 for n > 0.

cp's OEIS Frontend

A242598 Triangle read by rows: T(n,k) is the coefficient A_k in the transformation of 1 + x + x^2 + ... + x^n to the polynomial A_k*(x-k)^k for 0 <= k <= n.

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A248829 Triangle read by rows: T(n,k) is the coefficient A_k in the transformation of 1 + x + x^2 + ... + x^n to the polynomial A_k*(x+2k)^k for 0 <= k <= n .

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A248977 Triangle read by rows: T(n,k) is the coefficient A_k in the transformation of 1 + x + x^2 + ... + x^n to the polynomial A_k*(x+3k)^k for 0 <= k <= n.

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