A153656 -id:A153656 - cp's OEIS frontend

A153654 Triangle T(n, k, j) = T(n-1, k, j) + T(n-1, k-1, j) + (2j + 1)prime(j)*T(n-2, k-1, j) with T(2, k, j) = prime(j) and j = 9, read by rows.

2, 23, 23, 2, 1054, 2, 2, 12165, 12165, 2, 2, 13041, 484928, 13041, 2, 2, 13917, 5814074, 5814074, 13917, 2, 2, 14793, 11526908, 223541684, 11526908, 14793, 2, 2, 15669, 17623430, 2775818930, 2775818930, 17623430, 15669, 2, 2, 16545, 24103640, 7830701156, 103239353768, 7830701156, 24103640, 16545, 2

Offset: 1

Roger L. Bagula, Dec 30 2008

			Triangle begins as:
   2;
  23,    23;
   2,  1054,        2;
   2, 12165,    12165,          2;
   2, 13041,   484928,      13041,            2;
   2, 13917,  5814074,    5814074,        13917,          2;
   2, 14793, 11526908,  223541684,     11526908,      14793,        2;
   2, 15669, 17623430, 2775818930,   2775818930,   17623430,    15669,     2;
   2, 16545, 24103640, 7830701156, 103239353768, 7830701156, 24103640, 16545, 2;

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

Cf. A153652 (j=7), A153653 (j=8), this sequence (j=9), A153655 (j=10).

Cf. A153516, A153518, A153520, A153521, A153648, A153649, A153650, A153651, A153656, A153657.

f:= func< n,j | Round(((3-(-1)^n)/2)*NthPrime(j)^(n-1) - 2^((3-(-1)^n)/2)) >;
function T(n,k,j)
  if n eq 2 then return NthPrime(j);
  elif (n eq 3 and k eq 2 or n eq 4 and k eq 2 or n eq 4 and k eq 3) then return f(n,j);
  elif (k eq 1 or k eq n) then return 2;
  else return T(n-1,k,j) + T(n-1,k-1,j) + (2*j+1)*NthPrime(j)*T(n-2,k-1,j);
  end if; return T;
end function;
[T(n,k,9): k in [1..n], n in [1..12]]; // G. C. Greubel, Mar 03 2021

T[n_, k_, j_]:= T[n,k,j]= If[n==2, Prime[j], If[n==3 && k==2 || n==4 && 2<=k<=3, ((3-(-1)^n)/2)*Prime[j]^(n-1) -2^((3-(-1)^n)/2), If[k==1 || k==n, 2, T[n-1,k,j] + T[n-1,k-1,j] + (2*j+1)*Prime[j]*T[n-2,k-1,j] ]]];
Table[T[n,k,9], {n,12}, {k,n}]//Flatten (* modified by G. C. Greubel, Mar 03 2021 *)

@CachedFunction
def f(n,j): return ((3-(-1)^n)/2)*nth_prime(j)^(n-1) - 2^((3-(-1)^n)/2)
def T(n,k,j):
    if (n==2): return nth_prime(j)
    elif (n==3 and k==2 or n==4 and 2<=k<=3): return f(n,j)
    elif (k==1 or k==n): return 2
    else: return T(n-1,k,j) + T(n-1,k-1,j) + (2*j+1)*nth_prime(j)*T(n-2,k-1,j)
flatten([[T(n,k,9) for k in (1..n)] for n in (1..12)]) # G. C. Greubel, Mar 03 2021

T(n, k, j) =  T(n-1, k, j) + T(n-1, k-1, j) + (2*j + 1)*prime(j)*T(n-2, k-1, j) with T(2, k, j) = prime(j), T(3, 2, j) = 2*prime(j)^2 - 4, T(4, 2, j) = T(4, 3, j) = prime(j)^2 - 2, T(n, 1, j) = T(n, n, j) = 2 and j = 9.
From G. C. Greubel, Mar 04 2021: (Start)
Sum_{k=0..n} T(n, k, 10) = -(68/361)*[n=0] - (92/19)*[n=1] + 1058*(i*sqrt(437))^(n-2)*(ChebyshevU(n-2, -i/sqrt(437)) - (21*i/sqrt(437))*ChebyshevU(n-3, -i/sqrt(437) )).
Row sums satisfy the recurrence S(n) = 2*S(n-1) + 437*S(n-2) for n>4 with S(0) = 2, S(1) = 46, S(2) = 1058, S(3) = 24334. (End)

Edited by G. C. Greubel, Mar 03 2021

A153655 Triangle T(n, k, j) = T(n-1, k, j) + T(n-1, k-1, j) + (2j + 1)prime(j)*T(n-2, k-1, j) with T(2, k, j) = prime(j) and j = 10, read by rows.

Original entry on oeis.org

2, 29, 29, 2, 1678, 2, 2, 24387, 24387, 2, 2, 25607, 1070676, 25607, 2, 2, 26827, 15947966, 15947966, 26827, 2, 2, 28047, 31569456, 683937616, 31569456, 28047, 2, 2, 29267, 47935146, 10427818366, 10427818366, 47935146, 29267, 2, 2, 30487, 65045036, 29701552216, 437373644876, 29701552216, 65045036, 30487, 2

Offset: 1

Roger L. Bagula, Dec 30 2008

			Triangle begins as:
   2;
  29,    29;
   2,  1678,        2;
   2, 24387,    24387,           2;
   2, 25607,  1070676,       25607,            2;
   2, 26827, 15947966,    15947966,        26827,           2;
   2, 28047, 31569456,   683937616,     31569456,       28047,        2;
   2, 29267, 47935146, 10427818366,  10427818366,    47935146,    29267,     2;
   2, 30487, 65045036, 29701552216, 437373644876, 29701552216, 65045036, 30487, 2;

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

Cf. A153652 (j=7), A153653 (j=8), A153654 (j=9), this sequence (j=10).

Cf. A153516, A153518, A153520, A153521, A153648, A153649, A153650, A153651, A153656, A153657.

f:= func< n,j | Round(((3-(-1)^n)/2)*NthPrime(j)^(n-1) - 2^((3-(-1)^n)/2)) >;
function T(n,k,j)
  if n eq 2 then return NthPrime(j);
  elif (n eq 3 and k eq 2 or n eq 4 and k eq 2 or n eq 4 and k eq 3) then return f(n,j);
  elif (k eq 1 or k eq n) then return 2;
  else return T(n-1,k,j) + T(n-1,k-1,j) + (2*j+1)*NthPrime(j)*T(n-2,k-1,j);
  end if; return T;
end function;
[T(n,k,10): k in [1..n], n in [1..12]]; // G. C. Greubel, Mar 03 2021

T[n_, k_, j_]:= T[n,k,j]= If[n==2, Prime[j], If[n==3 && k==2 || n==4 && 2<=k<=3, ((3-(-1)^n)/2)*Prime[j]^(n-1) -2^((3-(-1)^n)/2), If[k==1 || k==n, 2, T[n-1,k,j] + T[n-1,k-1,j] + (2*j+1)*Prime[j]*T[n-2,k-1,j] ]]];
Table[T[n,k,10], {n,12}, {k,n}]//Flatten (* modified by G. C. Greubel, Mar 03 2021 *)

@CachedFunction
def f(n,j): return ((3-(-1)^n)/2)*nth_prime(j)^(n-1) - 2^((3-(-1)^n)/2)
def T(n,k,j):
    if (n==2): return nth_prime(j)
    elif (n==3 and k==2 or n==4 and 2<=k<=3): return f(n,j)
    elif (k==1 or k==n): return 2
    else: return T(n-1,k,j) + T(n-1,k-1,j) + (2*j+1)*nth_prime(j)*T(n-2,k-1,j)
flatten([[T(n,k,10) for k in (1..n)] for n in (1..12)]) # G. C. Greubel, Mar 03 2021

T(n, k, j) =  T(n-1, k, j) + T(n-1, k-1, j) + (2*j + 1)*prime(j)*T(n-2, k-1, j) with T(2, k, j) = prime(j), T(3, 2, j) = 2*prime(j)^2 - 4, T(4, 2, j) = T(4, 3, j) = prime(j)^2 - 2, T(n, 1, j) = T(n, n, j) = 2 and j = 10.
From G. C. Greubel, Mar 04 2021: (Start)
Sum_{k=0..n} T(n, k, 10) = -(76/147)*[n=0] - (116/7)*[n=1] + 1682*(i*sqrt(609))^(n-2)*(ChebyshevU(n-2, -i/sqrt(609)) - (27*i/sqrt(609))*ChebyshevU(n-3, -i/sqrt(609) )).
Row sums satisfy the recurrence S(n) = 2*S(n-1) + 609*S(n-2) for n>4 with S(0) = 2, S(1) = 58, S(2) = 1682, S(3) = 48778. (End)

Edited by G. C. Greubel, Mar 03 2021

A153657 Triangle T(n, k) = T(n-1, k) + T(n-1, k-1) + (2j +7)prime(j)*T(n-2, k-1) with j=10, read by rows.

Original entry on oeis.org

2, 29, 29, 2, 1678, 2, 2, 24387, 24387, 2, 2, 25955, 1362648, 25955, 2, 2, 27523, 20483624, 20483624, 27523, 2, 2, 29091, 40833912, 1107920632, 40833912, 29091, 2, 2, 30659, 62413512, 17187432136, 17187432136, 62413512, 30659, 2, 2, 32227, 85222424, 49222798744, 901876719128, 49222798744, 85222424, 32227, 2

Offset: 1

Roger L. Bagula, Dec 30 2008

			Triangle begins as:
   2;
  29,    29;
   2,  1678,        2;
   2, 24387,    24387,           2;
   2, 25955,  1362648,       25955,            2;
   2, 27523, 20483624,    20483624,        27523,           2;
   2, 29091, 40833912,  1107920632,     40833912,       29091,       2;
   2, 30659, 62413512, 17187432136,  17187432136,    62413512,    30659,     2;
   2, 32227, 85222424, 49222798744, 901876719128, 49222798744, 85222424, 32227, 2;

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

Sequences with variable (p,q,j): A153516 (0,1,2), A153518 (0,1,3), A153520 (0,1,4), A153521 (0,1,5), A153648 (1,0,3), A153649 (1,1,4), A153650 (1,4,5), A153651 (1,5,6), A153652 (2,1,7), A153653 (2,1,8), A153654 (2,1,9), A153655 (2,1,10), A153656 (2,3,9), this sequence (2,7,10).

Cf. A009973 (powers of 29).

f:= func< n,j | Round(((3-(-1)^n)/2)*NthPrime(j)^(n-1) - 2^((3-(-1)^n)/2)) >;
function T(n,k,p,q,j)
  if n eq 2 then return NthPrime(j);
  elif (n eq 3 and k eq 2 or n eq 4 and k eq 2 or n eq 4 and k eq 3) then return f(n,j);
  elif (k eq 1 or k eq n) then return 2;
  else return T(n-1,k,p,q,j) + T(n-1,k-1,p,q,j) + (p*j+q)*NthPrime(j)*T(n-2,k-1,p,q,j);
  end if; return T;
end function;
[T(n,k,2,7,10): k in [1..n], n in [1..12]]; // G. C. Greubel, Mar 06 2021

T[n_, k_, p_, q_, j_]:= T[n,k,p,q,j]= If[n==2, Prime[j], If[n==3 && k==2 || n==4 && 2<=k<=3, ((3-(-1)^n)/2)*Prime[j]^(n-1) -2^((3-(-1)^n)/2), If[k==1 || k==n, 2, T[n-1,k,p,q,j] + T[n-1,k-1,p,q,j] + (p*j+q)*Prime[j]*T[n-2,k-1,p,q,j] ]]];
Table[T[n,k,2,7,10], {n,12}, {k,n}]//Flatten (* modified by G. C. Greubel, Mar 06 2021 *)

@CachedFunction
def f(n,j): return ((3-(-1)^n)/2)*nth_prime(j)^(n-1) - 2^((3-(-1)^n)/2)
def T(n,k,p,q,j):
    if (n==2): return nth_prime(j)
    elif (n==3 and k==2 or n==4 and 2<=k<=3): return f(n,j)
    elif (k==1 or k==n): return 2
    else: return T(n-1,k,2,7,10) + T(n-1,k-1,p,q,j) + (p*j+q)*nth_prime(j)*T(n-2,k-1,p,q,j)
flatten([[T(n,k,p,q,j) for k in (1..n)] for n in (1..12)]) # G. C. Greubel, Mar 06 2021

T(n, k) = T(n-1, k) + T(n-1, k-1) + (2*j +7)*prime(j)*T(n-2, k-1) with j=10.
From G. C. Greubel, Mar 06 2021: (Start)
T(n,k,p,q,j) = T(n-1,k,p,q,j) + T(n-1,k-1,p,q,j) + (p*j+q)*prime(j)*T(n-2,k-1,p,q,j) with T(2,k,p,q,j) = prime(j), T(3,2,p,q,j) = 2*prime(j)^2 -4, T(4,2,p,q,j) = T(4,3,p,q,j) = prime(j)^2 -2, T(n,1,p,q,j) = T(n,n,p,q,j) = 2 and (p,q,j) = (2,7,10).
Sum_{k=0..n} T(n,k,p,q,j) = 2*prime(j)^(n-1), for (p,q,j)=(2,7,10), = 2*A009973(n-1). (End)

Edited by G. C. Greubel, Mar 06 2021

cp's OEIS Frontend

A153654 Triangle T(n, k, j) = T(n-1, k, j) + T(n-1, k-1, j) + (2j + 1)prime(j)*T(n-2, k-1, j) with T(2, k, j) = prime(j) and j = 9, read by rows.

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Magma

Mathematica

Sage

Formula

Extensions

A153655 Triangle T(n, k, j) = T(n-1, k, j) + T(n-1, k-1, j) + (2j + 1)prime(j)*T(n-2, k-1, j) with T(2, k, j) = prime(j) and j = 10, read by rows.

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Magma

Mathematica

Sage

Formula

Extensions

A153657 Triangle T(n, k) = T(n-1, k) + T(n-1, k-1) + (2j +7)prime(j)*T(n-2, k-1) with j=10, read by rows.

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Magma

Mathematica

Sage

Formula

Extensions

cp's OEIS Frontend

A153654 Triangle T(n, k, j) = T(n-1, k, j) + T(n-1, k-1, j) + (2*j + 1)*prime(j)*T(n-2, k-1, j) with T(2, k, j) = prime(j) and j = 9, read by rows.

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Magma

Mathematica

Sage

Formula

Extensions

A153655 Triangle T(n, k, j) = T(n-1, k, j) + T(n-1, k-1, j) + (2*j + 1)*prime(j)*T(n-2, k-1, j) with T(2, k, j) = prime(j) and j = 10, read by rows.

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Magma

Mathematica

Sage

Formula

Extensions

A153657 Triangle T(n, k) = T(n-1, k) + T(n-1, k-1) + (2*j +7)*prime(j)*T(n-2, k-1) with j=10, read by rows.

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Magma

Mathematica

Sage

Formula

Extensions

A153654 Triangle T(n, k, j) = T(n-1, k, j) + T(n-1, k-1, j) + (2j + 1)prime(j)*T(n-2, k-1, j) with T(2, k, j) = prime(j) and j = 9, read by rows.

A153655 Triangle T(n, k, j) = T(n-1, k, j) + T(n-1, k-1, j) + (2j + 1)prime(j)*T(n-2, k-1, j) with T(2, k, j) = prime(j) and j = 10, read by rows.

A153657 Triangle T(n, k) = T(n-1, k) + T(n-1, k-1) + (2j +7)prime(j)*T(n-2, k-1) with j=10, read by rows.