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.
Original entry on oeis.org
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
Offset: 0
Triangle begins as:
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;
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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 *)
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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
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.
Original entry on oeis.org
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
Offset: 0
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;
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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 *)
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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
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.
Original entry on oeis.org
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
Offset: 1
Triangle begins:
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;
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[ 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
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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
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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
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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
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row(n) = Vec(7*(1 - x)^(n+1)*polylog(-n,x)/x - 6*(1 + x)^(n-1)); \\ Michel Marcus, Jun 08 2018
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[[ 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
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