cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

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A167884 Triangle read by rows: T(n,k) given by T(n, 1) = T(n,n) = 1, otherwise T(n, k) = (m*n-m*k+1)*T(n-1,k-1) + (m*k-m+1)*T(n-1,k), where m = 8.

Original entry on oeis.org

1, 1, 1, 1, 18, 1, 1, 179, 179, 1, 1, 1636, 6086, 1636, 1, 1, 14757, 144362, 144362, 14757, 1, 1, 132854, 2941135, 7218100, 2941135, 132854, 1, 1, 1195735, 55446309, 277509955, 277509955, 55446309, 1195735, 1, 1, 10761672, 1001178268, 9211047544, 18315657030, 9211047544, 1001178268, 10761672, 1
Offset: 1

Views

Author

Roger L. Bagula, Nov 14 2009

Keywords

Examples

			Triangle begins as:
  1;
  1,       1;
  1,      18,        1;
  1,     179,      179,         1;
  1,    1636,     6086,      1636,         1;
  1,   14757,   144362,    144362,     14757,        1;
  1,  132854,  2941135,   7218100,   2941135,   132854,       1;
  1, 1195735, 55446309, 277509955, 277509955, 55446309, 1195735, 1;

Links

Crossrefs

For m = ...,-2,-1,0,1,2,3,4,5,6,7,8, ... we get ..., A225372, A144431, A007318, A008292, A060187, A142458, A142459, A142460, A142461, A142462, A167884, ...
Cf. A084948 (row sums).

Programs

  • Mathematica
    T[n_, k_, m_]:= T[n, k, m]= If[k==1 || k==n, 1, (m*n-m*k+1)*T[n-1, k-1, m] + (m*k-m+1)*T[n-1, k, m]];
A167884[n_, k_]:= T[n,k,8];
Table[A167884[n, k], {n,12}, {k,n}]//Flatten (* modified by G. C. Greubel, Mar 18 2022 *)
  • Sage
    @CachedFunction
def T(n,k,m):
    if (k==1 or k==n): return 1
    else: return (m*(n-k)+1)*T(n-1,k-1,m) + (m*k-m+1)*T(n-1,k,m)
def A167884(n,k): return T(n,k,8)
flatten([[ A167884(n,k) for k in (1..n)] for n in (1..15)]) # G. C. Greubel, Mar 18 2022

Formula

T(n, k) = (m*n-m*k+1)*T(n-1,k-1) + (m*k-m+1)*T(n-1,k), with T(n, 1) = T(n, n) = 1, and m = 8.
Sum_{k=1..n} T(n, k) = A084948(n-1).

Extensions

Edited by N. J. A. Sloane, May 08 2013

A225372 Triangle read by rows: T(n,k) (1 <= k <= n) given by T(n, 1) = T(n,n) = 1, otherwise T(n, k) = (m*n-m*k+1)*T(n-1,k-1) + (m*k-m+1)*T(n-1,k), where m = -2.

Original entry on oeis.org

1, 1, 1, 1, -2, 1, 1, -1, -1, 1, 1, -4, 6, -4, 1, 1, -3, 2, 2, -3, 1, 1, -6, 15, -20, 15, -6, 1, 1, -5, 9, -5, -5, 9, -5, 1, 1, -8, 28, -56, 70, -56, 28, -8, 1, 1, -7, 20, -28, 14, 14, -28, 20, -7, 1, 1, -10, 45, -120, 210, -252, 210, -120, 45, -10, 1
Offset: 1

Views

Author

N. J. A. Sloane and Roger L. Bagula, May 08 2013

Keywords

Examples

			Triangle begins:
  1;
  1,  1;
  1, -2,  1;
  1, -1, -1,   1;
  1, -4,  6,  -4,  1;
  1, -3,  2,   2, -3,   1;
  1, -6, 15, -20, 15,  -6,   1;
  1, -5,  9,  -5, -5,   9,  -5,  1;
  1, -8, 28, -56, 70, -56,  28, -8,  1;
  1, -7, 20, -28, 14,  14, -28, 20, -7, 1;

Links

Crossrefs

For m = ...,-2,-1,0,1,2,3,4,5,6,7,8, ... we get ..., A225372, A144431, A007318, A008292, A060187, A142458, A142459, A142560, A142561, A142562, A167884, ...
Cf. A130706 (row sums).

Programs

  • Magma
    function T(n,k,m)
  if k eq 1 or k eq n then return 1;
  else return (m*(n-k)+1)*T(n-1,k-1,m) + (m*k-m+1)*T(n-1,k,m);
  end if; return T;
end function;
A225372:= func< n,k | T(n,k,-2) >;
[A225372(n,k): k in [1..n], n in [1..12]]; // G. C. Greubel, Mar 17 2022
  • Maple
    T:=proc(n,k,l) option remember;
if (n=1 or k=1 or k=n) then 1 else
(l*n-l*k+1)*T(n-1,k-1,l)+(l*k-l+1)*T(n-1,k,l); fi; end;
for n from 1 to 14 do lprint([seq(T(n,k,-2),k=1..n)]); od;
  • Mathematica
    T[n_, k_, l_] := T[n, k, l] = If[n == 1 || k == 1 || k == n, 1, (l*n-l*k+1)*T[n-1, k-1, l]+(l*k-l+1)*T[n-1, k, l]]; Table[T[n, k, -2], {n, 1, 14}, {k, 1, n}] // Flatten (* Jean-François Alcover, Jan 09 2014, translated from Maple *)
  • Sage
    @CachedFunction
def T(n,k,m):
    if (k==1 or k==n): return 1
    else: return (m*(n-k)+1)*T(n-1,k-1,m) + (m*k-m+1)*T(n-1,k,m)
def A225372(n,k): return T(n,k,-2)
flatten([[ A225372(n,k) for k in (1..n)] for n in (1..15)]) # G. C. Greubel, Mar 17 2022

Formula

T(n, k) = (m*n-m*k+1)*T(n-1,k-1) + (m*k-m+1)*T(n-1,k), with T(n, 1) = T(n, n) = 1, and m = -2.
Sum_{k=1..n} T(n, k) = A130706(n-1). - G. C. Greubel, Mar 17 2022

A257626 Triangle read by rows: T(n,k) = t(n-k, k); t(n,m) = f(m)*t(n-1,m) + f(n)*t(n,m-1), where f(x) = 3*x + 6.

Original entry on oeis.org

1, 6, 6, 36, 108, 36, 216, 1404, 1404, 216, 1296, 15876, 33696, 15876, 1296, 7776, 166212, 642492, 642492, 166212, 7776, 46656, 1659204, 10701720, 19274760, 10701720, 1659204, 46656, 279936, 16052580, 163263924, 481752360, 481752360, 163263924, 16052580, 279936
Offset: 0

Views

Author

Dale Gerdemann, May 10 2015

Keywords

Examples

			Triangle begins as:
       1;
       6,        6;
      36,      108,        36;
     216,     1404,      1404,       216;
    1296,    15876,     33696,     15876,      1296;
    7776,   166212,    642492,    642492,    166212,      7776;
   46656,  1659204,  10701720,  19274760,  10701720,   1659204,    46656;
  279936, 16052580, 163263924, 481752360, 481752360, 163263924, 16052580, 279936;

Links

Crossrefs

Cf. A051609 (row sums), A142458, A257610, A257620, A257622, A257624.
See similar sequences listed in A256890.

Programs

  • Mathematica
    T[n_, k_, a_, b_]:= T[n, k, a, b]= If[k<0 || k>n, 0, If[n==0, 1, (a*(n-k)+b)*T[n-1, k-1, a, b] + (a*k+b)*T[n-1, k, a, b]]];
Table[T[n,k,3,6], {n,0,12}, {k,0,n}]//Flatten (* G. C. Greubel, Mar 20 2022 *)
  • Sage
    def T(n,k,a,b): # A257626
    if (k<0 or k>n): return 0
    elif (n==0): return 1
    else: return  (a*k+b)*T(n-1,k,a,b) + (a*(n-k)+b)*T(n-1,k-1,a,b)
flatten([[T(n,k,3,6) for k in (0..n)] for n in (0..12)]) # G. C. Greubel, Mar 20 2022

Formula

T(n,k) = t(n-k, k); t(0,0) = 1, t(n,m) = 0 if n < 0 or m < 0, else t(n,m) = f(m)*t(n-1,m) + f(n)*t(n,m-1), where f(x) = 3*x + 6.
Sum_{k=0..n} T(n, k) = A051609(n).
T(n, k) = (a*k + b)*T(n-1, k) + (a*(n-k) + b)*T(n-1, k-1), with T(n, 0) = 1, a = 3, and b = 6. - G. C. Greubel, Mar 20 2022

A142976 a(n) = (1/18)*(9*n^2 + 21*n + 10 - 4^(n+2)*(3*n+5) + 10*7^(n+1)).

Original entry on oeis.org

1, 39, 546, 5482, 47175, 373809, 2824048, 20729340, 149474205, 1065892555, 7547929806, 53215791774, 374165893891, 2626319535477, 18415017346620, 129036833755984, 903819045351033, 6329115592649775, 44313888005135290, 310239730485553170
Offset: 1

Views

Author

Roger L. Bagula and Gary W. Adamson, Oct 01 2008

Keywords

Links

Crossrefs

Cf. A142458, A144380, A144381, A144414.

Programs

  • Magma
    [5/9 + n^2/2 + 7*n/6 - 4^(n+1) * (2*n/3 + 10/9) + 5*7^(n+1)/9: n in [1..25]]; // Wesley Ivan Hurt, Oct 17 2017
  • Maple
    A142976:=n->5/9 + n^2/2 + 7*n/6 - 4^(n+1) * (2*n/3 + 10/9) + 5*7^(n+1)/9: seq(A142976(n), n=1..25); # Wesley Ivan Hurt, Oct 17 2017
  • Mathematica
    (* First program *)
T[n_, k_, m_]:= T[n, k, m]= If[k==1 || k==n, 1, (m*n-m*k+1)*T[n-1,k-1,m] + (m*k - m+1)*T[n-1,k,m]];
A142976[n_]:= T[n+2,n,3];
Table[A142976[n], {n,30}] (* modified by G. C. Greubel, Mar 16 2022 *)
(* Additional programs *)
CoefficientList[Series[(1 +21*x -36*x^2 -40*x^3)/((1-7*x)*(1-4*x)^2*(1-x)^3), {x, 0, 25}], x] (* Wesley Ivan Hurt, Oct 17 2017 *)
LinearRecurrence[{18,-120,374,-567,408,-112}, {1,39,546,5482,47175,373809}, 40] (* Vincenzo Librandi, Oct 18 2017 *)
  • Sage
    [(1/18)*(9*n^2 + 21*n + 10 - 4^(n+2)*(3*n+5) + 10*7^(n+1)) for n in (1..30)] # G. C. Greubel, Mar 16 2022

Formula

a(n) = A142458(n+2,n).
G.f.: x*(1+21*x-36*x^2-40*x^3) / ((1-7*x)*(4*x-1)^2*(1-x)^3). - R. J. Mathar, Sep 14 2013

A144414 a(n) = 2*(4^n - 1)/3 - n.

Original entry on oeis.org

1, 8, 39, 166, 677, 2724, 10915, 43682, 174753, 699040, 2796191, 11184798, 44739229, 178956956, 715827867, 2863311514, 11453246105, 45812984472, 183251937943, 733007751830, 2932031007381, 11728124029588, 46912496118419
Offset: 1

Views

Author

Roger L. Bagula and Gary W. Adamson, Oct 01 2008

Keywords

Links

Crossrefs

Cf. A142458, A142976, A144380.

Programs

  • Magma
    [(2^(2*n+1) -3*n -2)/3: n in [1..50]]; // G. C. Greubel, Mar 28 2021
  • Mathematica
    Table[2(4^n-1)/3 -n,{n,30}] (* or *) LinearRecurrence[{6,-9,4},{1,8,39},30] (* Harvey P. Dale, Mar 17 2015 *)
  • Sage
    [(2^(2*n+1) -3*n -2)/3 for n in (1..50)] # G. C. Greubel, Mar 28 2021

Formula

a(n) = A142458(n+1,n).
a(n) = A020988(n) - n. - R. J. Mathar, Nov 21 2008
G.f.: x*(1+2*x)/((1-x)^2*(1-4*x)). - Colin Barker, Jan 11 2012
a(1)=1, a(2)=8, a(3)=39, a(n) = 6*a(n-1) - 9*a(n-2) + 4*a(n-3). - Harvey P. Dale, Mar 17 2015
E.g.f.: (1/3)*(-2 - 3*x + 2*exp(x))*exp(x). - G. C. Greubel, Mar 28 2021

A166346 Coefficients of numerator of recursively defined rational function: p(x,3)=x*(x^2 + 8*x + 1)/(1 - x)^4; p(x, n) = 2*x*D[p(x, n - 1), x] - p(x,n-2).

Original entry on oeis.org

1, 1, 1, 1, 8, 1, 1, 39, 39, 1, 1, 158, 482, 158, 1, 1, 605, 4194, 4194, 605, 1, 1, 2276, 31047, 67752, 31047, 2276, 1, 1, 8515, 210609, 856075, 856075, 210609, 8515, 1, 1, 31802, 1356368, 9367974, 17194910, 9367974, 1356368, 31802, 1, 1, 118713
Offset: 1

Views

Author

Roger L. Bagula, Oct 12 2009

Keywords

Examples

			{1},
{1, 1},
{1, 8, 1},
{1, 39, 39, 1},
{1, 158, 482, 158, 1},
{1, 605, 4194, 4194, 605, 1},
{1, 2276, 31047, 67752, 31047, 2276, 1},
{1, 8515, 210609, 856075, 856075, 210609, 8515, 1},
{1, 31802, 1356368, 9367974, 17194910, 9367974, 1356368, 31802, 1},
{1, 118713, 8453460, 93489572, 285010254, 285010254, 93489572, 8453460, 118713, 1},
{1, 443072, 51564829, 876484896, 4159141218, 6855899968, 4159141218, 876484896, 51564829, 443072, 1}

References

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

Crossrefs

Cf. A123125, A142458.

Programs

  • Mathematica
    p[x_, 0] := 1/(1 - x);
p[x_, 1] := x/(1 - x)^2;
p[x_, 2] := x*(1 + x)/(1 - x)^3;
p[x_, 3] := x*(x^2 + 8*x + 1)/(1 - x)^4;
p[x_, n_] := p[x, n] = 2*x*D[p[x, n - 1], x] - p[x, n - 2]
a = Table[CoefficientList[FullSimplify[ExpandAll[(1 - x)^(n + 1)*p[x, n]/x]], x], {n, 1, 11}];
Flatten[a]
Table[Apply[Plus, CoefficientList[FullSimplify[ExpandAll[(1 - x)^(n + 1)*p[x, n]/x]], x]], {n, 1, 11}];

Formula

p(x,0)= 1/(1 - x);
p(x,1)= x/(1 - x)^2;
p(x,2)= x*(1 + x)/(1 - x)^3;
p(x,3)= x*(x^2 +8*x + 1)/(1 - x)^4;
p(x,n)= 2*x*D[p[x, n - 1], x] - p[x, n - 2]

A257608 Triangle read by rows: T(n,k) = t(n-k, k); t(n,m) = f(m)*t(n-1,m) + f(n)*t(n,m-1), where f(x) = 9*x + 1.

Original entry on oeis.org

1, 1, 1, 1, 20, 1, 1, 219, 219, 1, 1, 2218, 8322, 2218, 1, 1, 22217, 220222, 220222, 22217, 1, 1, 222216, 5006247, 12332432, 5006247, 222216, 1, 1, 2222215, 105340629, 530539235, 530539235, 105340629, 2222215, 1, 1, 22222214, 2123693776, 19700767514, 39259903390, 19700767514, 2123693776, 22222214, 1
Offset: 0

Views

Author

Dale Gerdemann, May 03 2015

Keywords

Examples

			Triangle begins as:
  1;
  1,       1;
  1,      20,         1;
  1,     219,       219,         1;
  1,    2218,      8322,      2218,         1;
  1,   22217,    220222,    220222,     22217,         1;
  1,  222216,   5006247,  12332432,   5006247,    222216,       1;
  1, 2222215, 105340629, 530539235, 530539235, 105340629, 2222215, 1;

Links

Crossrefs

Cf. A084949 (row sums), A257619.
Cf. A007318, A008292, A060187, A142458, A142459, A142460, A142461, A142462, A144431, A167884
Similar sequences listed in A256890.

Programs

  • Mathematica
    T[n_, k_, a_, b_]:= T[n, k, a, b]= If[k<0 || k>n, 0, If[k==0 || k==n, 1, (a*(n-k)+b)*T[n-1, k-1, a, b] + (a*k+b)*T[n-1, k, a, b]]];
Table[T[n,k,9,1], {n,0,12}, {k,0,n}]//Flatten (* G. C. Greubel, Mar 20 2022 *)
  • Sage
    def T(n,k,a,b): # A257608
    if (k<0 or k>n): return 0
    elif (k==0 or k==n): return 1
    else: return  (a*k+b)*T(n-1,k,a,b) + (a*(n-k)+b)*T(n-1,k-1,a,b)
flatten([[T(n,k,9,1) for k in (0..n)] for n in (0..12)]) # G. C. Greubel, Mar 20 2022

Formula

T(n, k) = t(n-k, k), where t(n,k) = f(k)*t(n-1, k) + f(n)*t(n, k-1), and f(n) = 9*n + 1.
Sum_{k=0..n} T(n, k) = A084949(n).
T(n, k) = (a*k + b)*T(n-1, k) + (a*(n-k) + b)*T(n-1, k-1), with T(n, 0) = T(n, n) = 1, a = 9, and b = 1. - G. C. Greubel, Mar 20 2022

A168524 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 = -2, b = 2, c = 1.

Original entry on oeis.org

1, 1, 1, 1, 10, 1, 1, 39, 39, 1, 1, 120, 350, 120, 1, 1, 341, 2266, 2266, 341, 1, 1, 950, 12895, 28340, 12895, 950, 1, 1, 2659, 69201, 290891, 290891, 69201, 2659, 1, 1, 7540, 360772, 2661644, 4987254, 2661644, 360772, 7540, 1, 1, 21681, 1851948, 22618188, 72033750, 72033750, 22618188, 1851948, 21681, 1
Offset: 0

Views

Author

Roger L. Bagula, Nov 28 2009

Keywords

Examples

			Triangle of coefficients begins as:
  1;
  1,     1;
  1,    10,       1;
  1,    39,      39,        1;
  1,   120,     350,      120,        1;
  1,   341,    2266,     2266,      341,        1;
  1,   950,   12895,    28340,    12895,      950,        1;
  1,  2659,   69201,   290891,   290891,    69201,     2659,       1;
  1,  7540,  360772,  2661644,  4987254,  2661644,   360772,    7540,     1;
  1, 21681, 1851948, 22618188, 72033750, 72033750, 22618188, 1851948, 21681, 1;

Links

Crossrefs

Cf. A168523, A168525.
Cf. A142458, A142459.

Programs

  • 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, -2, 2, 1], {n, 0, 12}]//Flatten (* modified by G. C. Greubel, Mar 19 2022 *)
  • 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,-2,2,1) for k in (0..n)] for n in (0..m)]) # G. C. Greubel, Mar 19 2022

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 = -2, b = 2, c = 1.
T(n, n-k) = T(n, k). (End)

Extensions

Edited by G. C. Greubel, Mar 19 2022

A225398 Triangle read by rows: absolute values of odd-numbered rows of A225433.

Original entry on oeis.org

1, 1, 38, 1, 1, 676, 4806, 676, 1, 1, 10914, 362895, 1346780, 362895, 10914, 1, 1, 174752, 20554588, 263879264, 683233990, 263879264, 20554588, 174752, 1, 1, 2796190, 1063096365, 35677598760, 267248150610, 554291429748, 267248150610, 35677598760, 1063096365, 2796190, 1
Offset: 1

Views

Author

Roger L. Bagula, Apr 26 2013 (Entered by N. J. A. Sloane, May 06 2013)

Keywords

Examples

			Triangle begins:
  1;
  1,     38,        1;
  1,    676,     4806,       676,         1;
  1,  10914,   362895,   1346780,    362895,     10914,        1;
  1, 174752, 20554588, 263879264, 683233990, 263879264, 20554588, 174752, 1;

Links

Crossrefs

Cf. A034870, A171692, A225076.

Programs

  • Mathematica
    (* First program *)
t[n_, k_, m_]:= t[n,k,m]= If[k==1 || k==n, 1, (m*n-m*k+1)*t[n-1,k-1,m] + (m*k-(m- 1))*t[n-1,k,m]];
T[n_, k_]:= T[n, k]= t[n+1, k+1, 3]; (* t(n,k,3) = A142458 *)
Flatten[Table[CoefficientList[Sum[T[n, k]*x^k, {k,0,n}]/(1+x), x], {n, 1, 14, 2}]]
(* Second program *)
t[n_, k_, m_]:= t[n, k, m]= If[k==1 || k==n, 1, (m*n-m*k+1)*t[n-1,k-1,m] + (m*k-m +1)*t[n-1,k,m]]; (* t(n,k,3) = A142458 *)
A225398[n_, k_]:= A225398[n, k]= Sum[(-1)^(k-j-1)*t[2*n,j+1,3], {j,0,k-1}];
Table[A225398[n, k], {n,12}, {k,2*n-1}] //Flatten (* G. C. Greubel, Mar 19 2022 *)
  • Sage
    @CachedFunction
def T(n, k, m):
    if (k==1 or k==n): return 1
    else: return (m*(n-k)+1)*T(n-1, k-1, m) + (m*k-m+1)*T(n-1, k, m)
def A142458(n, k): return T(n, k, 3)
def A225398(n,k): return sum( (-1)^(k-j-1)*A142458(2*n, j+1) for j in (0..k-1) )
flatten([[A225398(n, k) for k in (1..2*n-1)] for n in (1..12)]) # G. C. Greubel, Mar 19 2022

Formula

From G. C. Greubel, Mar 19 2022: (Start)
T(n, k) = Sum_{j=0..k-1} (-1)^(k-j-1)*A142458(2*n, j+1).
T(n, n-k) = T(n, k). (End)

Extensions

Edited by N. J. A. Sloane, May 11 2013

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

Views

Author

Roger L. Bagula, Nov 28 2009

Keywords

Examples

			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;

Links

Crossrefs

Cf. A142458, A142459.
Cf. A168524, A168525.

Programs

  • 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 *)
  • 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

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)

Extensions

Edited by G. C. Greubel, Mar 19 2022
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