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A166345 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+2*x+x^2)/(1-x)^4, read by rows. 5
1, 1, 1, 1, 2, 1, 1, 7, 7, 1, 1, 18, 42, 18, 1, 1, 41, 198, 198, 41, 1, 1, 88, 799, 1584, 799, 88, 1, 1, 183, 2925, 10331, 10331, 2925, 183, 1, 1, 374, 10056, 58874, 103310, 58874, 10056, 374, 1, 1, 757, 33160, 305888, 869794, 869794, 305888, 33160, 757, 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+2*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)*A005900(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, 2, 1;

1, 7, 7, 1;

1, 18, 42, 18, 1;

1, 41, 198, 198, 41, 1;

1, 88, 799, 1584, 799, 88, 1;

1, 183, 2925, 10331, 10331, 2925, 183, 1;

1, 374, 10056, 58874, 103310, 58874, 10056, 374, 1;

1, 757, 33160, 305888, 869794, 869794, 305888, 33160, 757, 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+10*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, m]= 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, -1], {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 A166345(n, k): return 1 if (k==1) else t(n-1, k, -1) - t(n-1, k-1, -1)

flatten([[A166345(n, k) for k in (1..n)] for n in (1..12)]) # G. C. Greubel, Mar 11 2022

CROSSREFS

Cf. A166340, A166341, A166343, A166344, A166346, A166349.

Cf. A005900, A123125.

Sequence in context: A303817 A158200 A220602 * A015110 A128596 A176305

Adjacent sequences: A166342 A166343 A166344 * A166346 A166347 A166348

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 December 1 15:29 EST 2022. Contains 358468 sequences. (Running on oeis4.)