A168524 -id:A168524

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A168523

Triangle of coefficients of g.f. a*(1+x)^n + b*(1-x)^(n+2)*polylog(-n-1, x)/x + 2^n*c*(1-x)^(n+1)*LerchPhi(x, -n, 1/2), with a = -1, b = 1, c = 1.

+10
3

1, 1, 1, 1, 8, 1, 1, 31, 31, 1, 1, 98, 290, 98, 1, 1, 289, 1974, 1974, 289, 1, 1, 836, 11719, 25944, 11719, 836, 1, 1, 2419, 64929, 275307, 275307, 64929, 2419, 1, 1, 7046, 346192, 2573466, 4831134, 2573466, 346192, 7046, 1, 1, 20677, 1804144, 22163080, 70723522, 70723522, 22163080, 1804144, 20677, 1

(list; table; graph; refs; listen; history; text; internal format)

OFFSET

0,5

LINKS

G. C. Greubel, Rows n = 0..50 of the triangle, flattened

FORMULA

From G. C. Greubel, Mar 19 2022: (Start)

G.f.: a*(1+x)^n + b*(1-x)^(n+2)*polylog(-n-1, x)/x + 2^n*c*(1-x)^(n+1)*LerchPhi(x, -n, 1/2), with a = -1, b = 1, c = 1.

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

EXAMPLE

Triangle begins as:

1, 1;

1, 8, 1;

1, 31, 31, 1;

1, 98, 290, 98, 1;

1, 289, 1974, 1974, 289, 1;

1, 836, 11719, 25944, 11719, 836, 1;

1, 2419, 64929, 275307, 275307, 64929, 2419, 1;

1, 7046, 346192, 2573466, 4831134, 2573466, 346192, 7046, 1;

1, 20677, 1804144, 22163080, 70723522, 70723522, 22163080, 1804144, 20677, 1;

MATHEMATICA

T[n_, a_, b_, c_]:= CoefficientList[Series[a*(1+x)^n + b*(1-x)^(n+2)* PolyLog[-n-1, x]/x + 2^n*c*(1-x)^(n+1)*LerchPhi[x, -n, 1/2], {x, 0, 30}], x];

Table[T[n, -1, 1, 1], {n, 0, 12}]//Flatten (* modified by G. C. Greubel, Mar 19 2022 *)

PROG

(Sage)

m=12

def LerchPhi(x, s, a): return sum( x^j/(j+a)^s for j in (0..3*m) )

def p(n, x, a, b, c): return a*(1+x)^n + b*(1-x)^(n+2)*polylog(-n-1, x)/x + 2^n*c*(1-x)^(n+1)*LerchPhi(x, -n, 1/2)

def T(n, k, a, b, c): return ( p(n, x, a, b, c) ).series(x, n+1).list()[k]

flatten([[T(n, k, -1, 1, 1) for k in (0..n)] for n in (0..m)]) # G. C. Greubel, Mar 19 2022

CROSSREFS

Cf. A142458, A142459.

Cf. A168524, A168525.

KEYWORD

nonn,tabl

AUTHOR

Roger L. Bagula, Nov 28 2009

EXTENSIONS

Edited by G. C. Greubel, Mar 19 2022

STATUS

approved

A168525

Triangle of coefficients of g.f. a*(1+x)^n + b*(1-x)^(n+2)*polylog(-n-1, x)/x + 2^n*c*(1-x)^(n+1)*LerchPhi(x, -n, 1/2), with a = 65/2, b = -162/2, c = 135/2.

+10
3

19, 19, 19, 19, 146, 19, 19, 759, 759, 19, 19, 3154, 10374, 3154, 19, 19, 11543, 89398, 89398, 11543, 19, 19, 39210, 615669, 1394444, 615669, 39210, 19, 19, 127303, 3747297, 16267301, 16267301, 3747297, 127303, 19, 19, 401858, 21201472, 160611806, 302914330, 160611806, 21201472, 401858, 19

(list; table; graph; refs; listen; history; text; internal format)

OFFSET

0,1

LINKS

G. C. Greubel, Rows n = 0..50 of the triangle, flattened

FORMULA

From G. C. Greubel, Mar 19 2022: (Start)

G.f.: a*(1+x)^n + b*(1-x)^(n+2)*polylog(-n-1, x)/x + 2^n*c*(1-x)^(n+1)*LerchPhi(x, -n, 1/2), with a = 65/2, b = -162/2, c = 135/2.

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

EXAMPLE

Triangle begins as:

19;

19, 19;

19, 146, 19;

19, 759, 759, 19;

19, 3154, 10374, 3154, 19;

19, 11543, 89398, 89398, 11543, 19;

19, 39210, 615669, 1394444, 615669, 39210, 19;

19, 127303, 3747297, 16267301, 16267301, 3747297, 127303, 19;

19, 401858, 21201472, 160611806, 302914330, 160611806, 21201472, 401858, 19;

MATHEMATICA

T[n_, a_, b_, c_]:= CoefficientList[Series[a*(1+x)^n + b*(1-x)^(n+2)* PolyLog[-n-1, x]/x + 2^n*c*(1-x)^(n+1)*LerchPhi[x, -n, 1/2], {x, 0, 30}], x];

Table[T[n, 65/2, -162/2, 135/2], {n, 0, 12}]//Flatten (* modified by G. C. Greubel, Mar 19 2022 *)

PROG

(Sage)

m=12

def LerchPhi(x, s, a): return sum( x^j/(j+a)^s for j in (0..3*m) )

def p(n, x, a, b, c): return a*(1+x)^n + b*(1-x)^(n+2)*polylog(-n-1, x)/x + 2^n*c*(1-x)^(n+1)*LerchPhi(x, -n, 1/2)

def T(n, k, a, b, c): return ( p(n, x, a, b, c) ).series(x, n+1).list()[k]

flatten([[T(n, k, 65/2, -162/2, 135/2) for k in (0..n)] for n in (0..m)]) # G. C. Greubel, Mar 19 2022

CROSSREFS

Cf. A142458, A142459.

Cf. A168523, A168524.

KEYWORD

nonn,tabl

AUTHOR

Roger L. Bagula, Nov 28 2009

EXTENSIONS

Edited by G. C. Greubel, Mar 19 2022

STATUS

approved

A141697

T(n,k) = (q*Sum_{j=0..k+1} (-1)^j*binomial(n+1, j)*(k+1-j)^n - p*binomial(n-1, k))/2 where p=12 and q=14.

+10
2

1, 1, 1, 1, 16, 1, 1, 59, 59, 1, 1, 158, 426, 158, 1, 1, 369, 2054, 2054, 369, 1, 1, 804, 8247, 16792, 8247, 804, 1, 1, 1687, 29925, 109123, 109123, 29925, 1687, 1, 1, 3466, 102088, 617302, 1092910, 617302, 102088, 3466, 1, 1, 7037, 334664, 3185840, 9171722, 9171722, 3185840, 334664, 7037, 1

(list; table; graph; refs; listen; history; text; internal format)

OFFSET

1,5

COMMENTS

Row n is made of coefficients from 7*(1 - x)^(n+1) * polylog(-n,x)/x - 6*(1 + x)^(n-1). - Thomas Baruchel, Jun 03 2018

LINKS

G. C. Greubel, Rows n = 1..100 of triangle, flattened

Thomas Baruchel, A conjectured formula for the polylogarithm of a negative integer order, Mathematics Stack Exchange question, Jun 04 2018.

FORMULA

p=12; q=14; T(n,k) = (q*Sum_{j=0..k+1} (-1)^j*binomial(n+1, j)*(k+1-j)^n - p*binomial(n-1, k))/2.

a(n) = 3*A168524(n) - 2*A154337(n). - Thomas Baruchel, Jun 08 2018

EXAMPLE

Triangle begins:

1, 1;

1, 16, 1;

1, 59, 59, 1;

1, 158, 426, 158, 1;

1, 369, 2054, 2054, 369, 1;

1, 804, 8247, 16792, 8247, 804, 1;

1, 1687, 29925, 109123, 109123, 29925, 1687, 1;

MAPLE

T:= proc(n, k): 7*add((-1)^j*binomial(n+1, j)*(k-j+1)^n, j = 0..k+1) - 6*binomial(n-1, k); end proc; seq(seq(T(n, k), k=0..n-1), n=1..10); # G. C. Greubel, Nov 13 2019

MATHEMATICA

i=12; l=14; Table[Table[(l*Sum[(-1)^j*Binomial[n+1, j](k+1-j)^n, {j, 0, k+1}] - i*Binomial[n-1, k])/2, {k, 0, n-1}], {n, 10}]//Flatten

PROG

(PARI) T(n, k) = 7*sum(j=0, k+1, (-1)^j*binomial(n+1, j)*(k-j+1)^n) - 6* binomial(n-1, k);

for(n=1, 10, for(k=0, n-1, print1(T(n, k), ", "))) \\ G. C. Greubel, Jun 03 2018

(PARI) row(n) = Vec(7*(1 - x)^(n+1)*polylog(-n, x)/x - 6*(1 + x)^(n-1)); \\ Michel Marcus, Jun 08 2018

(Magma) [ 7*(&+[(-1)^j*Binomial(n+1, j)*(k-j+1)^n: j in [0..k+1]]) - 6*Binomial(n-1, k): k in [0..n-1], n in [1..10]]; // G. C. Greubel, Nov 13 2019

(Sage) [[ 7*sum( (-1)^j*binomial(n+1, j)*(k-j+1)^n for j in (0..k+1)) - 6*binomial(n-1, k) for k in (0..n-1)] for n in (1..10)] # G. C. Greubel, Nov 13 2019

CROSSREFS

Cf. Eulerian numbers (A008292) and Pascal's triangle (A007318).

Cf. A141696.

KEYWORD

nonn,tabl,easy

AUTHOR

Roger L. Bagula, Sep 11 2008

EXTENSIONS

Edited by G. C. Greubel, Nov 13 2019

STATUS

approved

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