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A166340 Triangle T(n, k) = coefficients of ( t(n, x) ) where t(n, x) = (1-x)^(n+1)*p(n, x)/x, p(n, x) = x*D( p(n-1, x) ), with p(1, x) = x/(1-x)^2, p(2, x) = x*(1+x)/(1-x)^3, and p(3, x) = x*(1+8*x+x^2)/(1-x)^4, read by rows. 5
1, 1, 1, 1, 8, 1, 1, 19, 19, 1, 1, 42, 114, 42, 1, 1, 89, 510, 510, 89, 1, 1, 184, 1975, 4080, 1975, 184, 1, 1, 375, 7029, 26195, 26195, 7029, 375, 1, 1, 758, 23712, 146954, 261950, 146954, 23712, 758, 1, 1, 1525, 77200, 753800, 2191474, 2191474, 753800, 77200, 1525, 1 (list; table; graph; refs; listen; history; text; internal format)
OFFSET

1,5

REFERENCES

Douglas C. Montgomery and Lynwood A. Johnson, Forecasting and Time Series Analysis, MaGraw-Hill, New York, 1976, page 91

LINKS

G. C. Greubel, Rows n = 1..50 of the triangle, flattened

FORMULA

T(n, k) = coefficients of ( t(n, x) ) where t(n, x) = (1-x)^(n+1)*p(n, x)/x, p(n, x) = x*D( p(n-1, x) ), with p(1, x) = x/(1-x)^2, p(2, x) = x*(1+x)/(1-x)^3, and p(3, x) = x*(1+8*x+x^2)/(1-x)^4.

From G. C. Greubel, Mar 11 2022: (Start)

T(n, k) = t(n-1, k) - t(n-1, k-1), T(n,1) = 1, where t(n, k) = Sum_{j=0..k} (-1)^(k-j)*binomial(n+1, k-j)*b(n, j), b(n, k) = k^(n-2)*A004466(k), b(n, 0) = 1, and b(1, k) = 1.

T(n, n-k) = T(n, k). (End)

EXAMPLE

Triangle begins as:

  1;

  1,    1;

  1,    8,     1;

  1,   19,    19,      1;

  1,   42,   114,     42,       1;

  1,   89,   510,    510,      89,       1;

  1,  184,  1975,   4080,    1975,     184,      1;

  1,  375,  7029,  26195,   26195,    7029,    375,     1;

  1,  758, 23712, 146954,  261950,  146954,  23712,   758,    1;

  1, 1525, 77200, 753800, 2191474, 2191474, 753800, 77200, 1525, 1;

MATHEMATICA

(* First program *)

p[x_, 1]:= x/(1-x)^2;

p[x_, 2]:= x*(1+x)/(1-x)^3;

p[x_, 3]:= x*(1+8*x+x^2)/(1-x)^4;

p[x_, n_]:= p[x, n]= x*D[p[x, n-1], x]

Table[CoefficientList[(1-x)^(n+1)*p[x, n]/x, x], {n, 12}]//Flatten

(* Second program *)

b[n_, k_, m_]:= If[n<2, 1, If[k==0, 0, k^(n-1)*((m+3)*k^2 - m)/3]];

t[n_, k_, m_]:= t[n, k]= Sum[(-1)^(k-j)*Binomial[n+1, k-j]*b[n, j, m], {j, 0, k}];

T[n_, k_, m_]:= T[n, k, m]= If[k==1, 1, t[n-1, k, m] - t[n-1, k-1, m]];

Table[T[n, k, 2], {n, 12}, {k, n}]//Flatten (* G. C. Greubel, Mar 11 2022 *)

PROG

(Sage)

def b(n, k, m):

    if (n<2): return 1

    elif (k==0): return 0

    else: return k^(n-1)*((m+3)*k^2 - m)/3

@CachedFunction

def t(n, k, m): return sum( (-1)^(k-j)*binomial(n+1, k-j)*b(n, j, m) for j in (0..k) )

def A166340(n, k): return 1 if (k==1) else t(n-1, k, 2) - t(n-1, k-1, 2)

flatten([[A166340(n, k) for k in (1..n)] for n in (1..12)]) # G. C. Greubel, Mar 11 2022

CROSSREFS

Cf. A166341, A166343, A166344, A166345, A166346, A166349.

Cf. A004466, A123125.

Sequence in context: A051469 A155494 A154227 * A157178 A174303 A176488

Adjacent sequences:  A166337 A166338 A166339 * A166341 A166342 A166343

KEYWORD

nonn,tabl

AUTHOR

Roger L. Bagula, Oct 12 2009

EXTENSIONS

Edited by G. C. Greubel, Mar 11 2022

STATUS

approved

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Last modified May 22 04:12 EDT 2022. Contains 353933 sequences. (Running on oeis4.)