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.

Showing 1-2 of 2 results.

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

A159041 Triangle read by rows: row n (n>=0) gives the coefficients of the polynomial p(n,x) of degree n defined in comments.

Original entry on oeis.org

1, 1, 1, 1, -10, 1, 1, -25, -25, 1, 1, -56, 246, -56, 1, 1, -119, 1072, 1072, -119, 1, 1, -246, 4047, -11572, 4047, -246, 1, 1, -501, 14107, -74127, -74127, 14107, -501, 1, 1, -1012, 46828, -408364, 901990, -408364, 46828, -1012, 1, 1, -2035, 150602, -2052886, 7685228, 7685228, -2052886, 150602, -2035, 1
Offset: 0

Views

Author

Roger L. Bagula, Apr 03 2009

Keywords

Comments

Let E(n,k) (1 <= k <= n) denote the Eulerian numbers as defined in A008292. Then we define polynomials p(n,x) for n >= 0 as follows.
p(n,x) = (1/(1-x)) * ( Sum_{k=0..floor(n/2)} (-1)^k*E(n+2,k+1)*x^k + Sum_{k=ceiling((n+2)/2)..n+1} (-1)^(n+k)*E(n+2,k+1)*x^k ).
For example,
p(0,x) = (1-x)/(1-x) = 1,
p(1,x) = (1-x^2)/(1-x) = 1 + x,
p(2,x) = (1 - 11*x + 11*x^2 - x^3)/(1-x) = 1 - 10*x + x^2,
p(3,x) = (1 - 26*x + 26*x^3 - x^4)/(1-x) = 1 - 25*x - 25*x^2 + x^3,
p(4,x) = (1 - 57*x + 302*x^2 - 302*x^3 + 57*x^3 + x^5)/(1-x)
= 1 - 56*x + 246*x^2 - 56*x^3 + x^4.
More generally, there is a triangle-to-triangle transformation U -> T defined as follows.
Let U(n,k) (1 <= k <= n) be a triangle of nonnegative numbers in which the rows are symmetric about the middle. Define polynomials p(n,x) for n >= 0 by
p(n,x) = (1/(1-x)) * ( Sum_{k=0..floor(n/2)} (-1)^k*U(n+2,k+1)*x^k + Sum_{k=ceiling((n+2)/2)..n+1} (-1)^(n+k)*U(n+2,k+1)*x^k ).
The n-th row of the new triangle T(n,k) (0 <= k <= n) gives the coefficients in the expansion of p(n+2).
The new triangle may be defined recursively by: T(n,0)=1; T(n,k) = T(n,k-1) + (-1)^k*U(n+2,k) for 1 <= k <= floor(n/2); T(n,k) = T(n,n-k).
Note that the central terms in the odd-numbered rows of U(n,k) do not get used.
The following table lists various sequences constructed using this transform:
Parameter Triangle Triangle Odd-numbered
m U T rows
0 A007312 A007312 A034870
1 A008292 A159041 A171692
2 A060187 A225356 A225076
3 A142458 A225433 A225398
4 A142459 A225434 A225415

Examples

			Triangle begins as follows:
  1;
  1,     1;
  1,   -10,      1;
  1,   -25,    -25,        1;
  1,   -56,    246,      -56,       1;
  1,  -119,   1072,     1072,    -119,       1;
  1,  -246,   4047,   -11572,    4047,    -246,        1;
  1,  -501,  14107,   -74127,  -74127,   14107,     -501,      1;
  1, -1012,  46828,  -408364,  901990, -408364,    46828,  -1012,     1;
  1, -2035, 150602, -2052886, 7685228, 7685228, -2052886, 150602, -2035, 1;

Links

Crossrefs

Cf. A007312, A008292, A034870, A060187, A142458, A142459, A159041, A171692, A225076, A225356, A225398, A225415, A225433, A225434.

Programs

  • Maple
    A008292 := proc(n, k) option remember; if k < 1 or k > n then 0; elif k = 1 or k = n then 1; else k*procname(n-1, k)+(n-k+1)*procname(n-1, k-1) ; end if; end proc:
# row n of new triangle T(n,k) in terms of old triangle U(n,k):
p:=proc(n) local k; global U;
simplify( (1/(1-x)) * ( add((-1)^k*U(n+2,k+1)*x^k,k=0..floor(n/2)) + add((-1)^(n+k)*U(n+2,k+1)*x^k, k=ceil((n+2)/2)..n+1 )) );
end;
U:=A008292;
for n from 0 to 6 do lprint(simplify(p(n))); od: # N. J. A. Sloane, May 11 2013
A159041 := proc(n, k)
    if k = 0 then
        1;
    elif k <= floor(n/2) then
        A159041(n, k-1)+(-1)^k*A008292(n+2, k+1) ;
    else
        A159041(n, n-k) ;
    end if;
end proc: # R. J. Mathar, May 08 2013
  • Mathematica
    A[n_, 1] := 1;
A[n_, n_] := 1;
A[n_, k_] := (n - k + 1)A[n - 1, k - 1] + k A[n - 1, k];
p[x_, n_] = Sum[x^i*If[i == Floor[n/2] && Mod[n, 2] == 0, 0, If[i <= Floor[n/2], (-1)^i*A[n, i], -(-1)^(n - i)*A[n, i]]], {i, 0, n}]/(1 - x);
Table[CoefficientList[FullSimplify[p[x, n]], x], {n, 1, 11}];
Flatten[%]
  • Sage
    def A008292(n,k): return sum( (-1)^j*(k-j)^n*binomial(n+1,j) for j in (0..k) )
@CachedFunction
def T(n,k):
    if (k==0 or k==n): return 1
    elif (k <= (n//2)): return T(n,k-1) + (-1)^k*A008292(n+2,k+1)
    else: return T(n,n-k)
flatten([[T(n,k) for k in (0..n)] for n in (0..12)]) # G. C. Greubel, Mar 18 2022

Formula

T(n, k) = T(n, k-1) + (-1)^k*A008292(n+2, k+1) if k <= floor(n/2), otherwise T(n, n-k), with T(n, 0) = T(n, n) = 1. - R. J. Mathar, May 08 2013

Extensions

Edited by N. J. A. Sloane, May 07 2013, May 11 2013
Showing 1-2 of 2 results.

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

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