A248811 -id:A248811 - cp's OEIS frontend

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.

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.

A249268 Triangle read by rows: T(n,k) is the coefficient A_k in the transformation Sum_{k=0..n} x^k = Sum_{k=0..n} A_k(x+3(-1)^k)^k.

Original entry on oeis.org

1, 4, 1, -23, -5, 1, -320, -86, 10, 1, 4297, 1102, -152, -11, 1, 92020, 24187, -3122, -281, 16, 1, -1922207, -502151, 66133, 5659, -389, -17, 1, -55746464, -14601740, 1908316, 167254, -10784, -584, 22, 1, 1589338993, 415992316, -54490040, -4745234, 312406, 16048, -734, -23, 1

Offset: 0

Derek Orr, Oct 23 2014

Consider the transformation 1 + x + x^2 + x^3 + ... + x^n = A_0*(x+3)^0 + A_1*(x-3)^1 + A_2*(x+3)^2 + A_3*(x-3)^3 + ... + A_n*(x+3*(-1)^n)^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;
4,                  1;
-23,               -5,         1;
-320,             -86,        10,        1;
4297,            1102,      -152,      -11,      1;
92020,          24187,     -3122,     -281,     16,     1;
-1922207,     -502151,     66133,     5659,   -389,   -17,    1;
-55746464,  -14601740,   1908316,   167254, -10784,  -584,   22,   1;
1589338993, 415992316, -54490040, -4745234, 312406, 16048, -734, -23, 1;
...

Cf. A248975, A248977, A248811.

a(n, j, L)=if(j==n, return(1)); if(j!=n, return(1-sum(i=1, n-j, (-L)^i*(-1)^(i*j)*binomial(i+j, i)*a(n, i+j, L))))
for(n=0, 10, for(j=0, n, print1(a(n, j, 3), ", ")))

cp's OEIS Frontend

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.

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

PARI

Formula

A249268 Triangle read by rows: T(n,k) is the coefficient A_k in the transformation Sum_{k=0..n} x^k = Sum_{k=0..n} A_k(x+3(-1)^k)^k.

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

PARI

cp's OEIS Frontend

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.

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

PARI

Formula

A249268 Triangle read by rows: T(n,k) is the coefficient A_k in the transformation Sum_{k=0..n} x^k = Sum_{k=0..n} A_k*(x+3*(-1)^k)^k.

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

PARI

A249268 Triangle read by rows: T(n,k) is the coefficient A_k in the transformation Sum_{k=0..n} x^k = Sum_{k=0..n} A_k(x+3(-1)^k)^k.