A191008 -id:A191008 - cp's OEIS frontend

A248811 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+3)^k for 0 <= k <= n.

1, -2, 1, 7, -5, 1, -20, 22, -8, 1, 61, -86, 46, -11, 1, -182, 319, -224, 79, -14, 1, 547, -1139, 991, -461, 121, -17, 1, -1640, 3964, -4112, 2374, -824, 172, -20, 1, 4921, -13532, 16300, -11234, 4846, -1340, 232, -23, 1, -14762, 45517, -62432, 50002, -25772, 8866, -2036, 301, -26, 1, 44287, -151313, 232813, -212438, 127318, -52370, 14974, -2939, 379, -29, 1

Offset: 0

Derek Orr, Oct 14 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_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.

			       1;
      -2,       1;
       7,      -5,      1;
     -20,      22,     -8,       1;
      61,     -86,     46,     -11,      1;
    -182,     319,   -224,      79,    -14,      1;
     547,   -1139,    991,    -461,    121,    -17,     1;
   -1640,    3964,  -4112,    2374,   -824,    172,   -20,     1;
    4921,  -13532,  16300,  -11234,   4846,  -1340,   232,   -23,   1;
  -14762,   45517, -62432,   50002, -25772,   8866, -2036,   301, -26,   1;
   44287, -151313, 232813, -212438, 127318, -52370, 14974, -2939, 379, -29, 1;

G. C. Greubel, Rows n=0..100 of triangle, flattened

Cf. A193843, A077925, A191008, A014983.

[[(&+[(-3)^(j-k)*Binomial(j,k): j in [0..n]]): k in [0..n]]: n in [0..20]]; // G. C. Greubel, May 27 2018

T[n_, k_]:= Sum[(-3)^(j-k)*Binomial[j,k], {j,0,n}]; Table[T[n, k], {n, 0, 10}, {k, 0, n}]//Flatten (* G. C. Greubel, May 27 2018 *)

for(n=0,20,for(k=0,n,print1(sum(i=0,n,((-3)^(i-k)* binomial(i, k)) ),", ")))

T(n,n-1) = -3*n + 1 for n > 0.
T(n,0) = A014983(n+1).
T(n,1) = (-1)^(n+1)*A191008(n-1).
Row n sums to A077925(n).

A191010 a(n) = (n4^(n+1) + (64^(n+1)+(-1)^n)/5)/5.

Original entry on oeis.org

1, 7, 41, 215, 1065, 5079, 23593, 107479, 482345, 2139095, 9395241, 40936407, 177167401, 762356695, 3264175145, 13915694039, 59098749993, 250138895319, 1055531162665, 4442026976215, 18647717207081, 78109306037207, 326510972984361, 1362338887279575

Offset: 0

Edward Omey, Jun 16 2011

a(n) = 4^(n+1)*H(2^n)/5 with H(2^n) = n+(6+(-1)^n/4^(n+1))/5 = E(N(2^n)), where X, X(1), X(2),... denote random variables with pdf P(X = 1) = P(X = 4) = 1/5 and P(X = 2) = 3/5, N(x) is the first value of k such that X(1)*X(2)*...*X(k) > x and H(x)= E(N(x)).

Colin Barker, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (7,-8,-16).

Cf. A191008.

seq((n*4^(n+1) + (6*4^(n+1)+(-1)^n)/5)/5, n=0..50); # Robert Israel, May 03 2017

CoefficientList[Series[1/((1 + x) (1 - 4 x)^2), {x, 0, 23}], x] (* or *)
LinearRecurrence[{7, -8, -16}, {1, 7, 41}, 24] (* Michael De Vlieger, May 03 2017 *)

a(n)= (n*4^(n+1)+(6*4^(n+1)+(-1)^n)/5)/5; \\ Michel Marcus, Oct 16 2014

Vec(1 / ((1 + x)*(1 - 4*x)^2) + O(x^30)) \\ Colin Barker, May 03 2017

a(n) = (n*4^(n+1) + (6*4^(n+1)+(-1)^n)/5)/5.
From Colin Barker, May 03 2017: (Start)
G.f.: 1 / ((1 + x)*(1 - 4*x)^2).
a(n) = 7*a(n-1) - 8*a(n-2) - 16*a(n-3) for n>2.
(End)
E.g.f.: (80*x*exp(4*x)+24*exp(4*x)+exp(-x))/25. - Robert Israel, May 03 2017

Formula corrected and more terms from Michel Marcus, Oct 16 2014

Edited by M. F. Hasler, Oct 16 2014

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

A248811 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+3)^k for 0 <= k <= n.

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A191010 a(n) = (n4^(n+1) + (64^(n+1)+(-1)^n)/5)/5.

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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

A248811 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+3)^k for 0 <= k <= n.

Views

Author

Keywords

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Examples

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Crossrefs

Programs

Magma

Mathematica

PARI

Formula

A191010 a(n) = (n*4^(n+1) + (6*4^(n+1)+(-1)^n)/5)/5.

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Maple

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Extensions

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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PARI

Formula

A191010 a(n) = (n4^(n+1) + (64^(n+1)+(-1)^n)/5)/5.

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.