A225076 -id:A225076 - cp's OEIS frontend

A002671 a(n) = 4^n(2n+1)!.

1, 24, 1920, 322560, 92897280, 40874803200, 25505877196800, 21424936845312000, 23310331287699456000, 31888533201572855808000, 53572735778642397757440000, 108431217215972213061058560000

Offset: 0

N. J. A. Sloane and Simon Plouffe

From Sanjar Abrarov, Mar 30 2019: (Start)
There is a formula for numerical integration (see MATLAB Central file ID# 71037):
Integral_{x=0..1} f(x) dx = 2*Sum_{m=1..M} Sum_{n>=0} 1/((2*M)^(2*n + 1)*(2*n + 1)!)*f^(2*n)(x)|_x = (m - 1/2)/M, where the notation f^(2*n)(x)|_x = (m - 1/2)/M is the (2*n)-th derivative of the function f(x) at the points x = (m - 1/2)/M.
When we choose M = 1, then the corresponding coefficients are generated as 2*1/(2^(2*n + 1)*(2*n + 1)!) = 1/(4^n*(2*n + 1)!).
Therefore, this sequence also occurs in the denominator of the numerical integration formula at M = 1. (End)
From Peter Bala, Oct 03 2019: (Start)
Denominators in the expansion of 2*sinh(x/2) = x + x^3/24 + x^5/1920 + x^7/322560 + ....
If f(x) is a polynomial in x then the central difference f(x+1/2) - f(x-1/2) = 2*sinh(D/2)(f(x)) = D(f(x)) + (1/24)*D^3(f(x)) + (1/1920)*D^5(f(x)) + ..., where D denotes the differential operator d/dx. Formulas for higher central differences in terms of powers of the operator D can be obtained from the expansion of powers of the function 2*sinh(x/2). For example, the expansion (2*sinh(x/2))^2 = x^2 + (1/12)*x^4 + (1/360)*x^6 + .. leads to the second central difference formula f(x+1) - 2*f(x) + f(x-1) = D^2(f(x)) + (1/12)*D^4(f(x)) + (1/360)* D^6(f(x)) + .... See A002674. (End)

N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Delbert L. Johnson, Table of n, a(n) for n = 0..201
S. M. Abrarov and B. M. Quine, Array numerical integration by enhanced midpoint rule, MATLAB Central file ID #: 71037.
H. E. Salzer, Tables of coefficients for obtaining central differences from the derivatives, Journal of Mathematics and Physics (this journal is also called Studies in Applied Mathematics), 42 (1963), 162-165, plus several inserted tables. [Note that there is a mistake in the definition of this sequence on line 2 of page 164.]
H. E. Salzer, Annotated scanned copy of left side of Table I.
Eric Weisstein's World of Mathematics, Central Difference.
Index to divisibility sequences.

A bisection of A002866 and (apart from initial term) also a bisection of A007346.
Row sums of A225076. - Roger L. Bagula, Apr 27 2013
Cf. A002672, A002673, A002674, A002675, A002676, A002677.

a[n_] := 4^n*(2*n + 1)!; Array[a, 12, 0] (* Amiram Eldar, Apr 09 2022 *)

PARI
```
a(n)=4^n*(2*n+1)!
```

a(n) = 16^n * Pochhammer(1,n) * Pochhammer(3/2,n). - Roger L. Bagula, Apr 26 2013
From Amiram Eldar, Apr 09 2022: (Start)
Sum_{n>=0} 1/a(n) = 2*sinh(1/2).
Sum_{n>=0} (-1)^n/a(n) = 2*sin(1/2). (End)

More terms from Michael Somos

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

Roger L. Bagula, Apr 03 2009

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

			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;

G. C. Greubel, Rows n = 0..50 of the triangle, flattened
Roger L. Bagula, Another Mathematica program for A159041.

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

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

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[%]

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

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

Edited by N. J. A. Sloane, May 07 2013, May 11 2013

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

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

			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;

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

Cf. A034870, A171692, A225076.

(* 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 *)

@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

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)

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

A225415 Triangle read by rows: absolute values of odd-numbered rows of A225434.

Original entry on oeis.org

1, 1, 58, 1, 1, 1556, 12006, 1556, 1, 1, 39054, 1461615, 5647300, 1461615, 39054, 1, 1, 976552, 135028828, 1838120344, 4873361350, 1838120344, 135028828, 976552, 1, 1, 24414050, 11462721645, 414730580760, 3221733789330, 6783391017228, 3221733789330, 414730580760, 11462721645, 24414050, 1

Offset: 1

Roger L. Bagula, May 07 2013

			Triangle begins:
  1;
  1,     58,         1;
  1,   1556,     12006,       1556,          1;
  1,  39054,   1461615,    5647300,    1461615,      39054,         1;
  1, 976552, 135028828, 1838120344, 4873361350, 1838120344, 135028828, 976552, 1;

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

The m=4 triangle in the sequence A034870 (m=0), A171692 (m=1), A225076 (m=2), A225398 (m=3).

(* 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+1,4]; (* t(n,k,4) = A142459 *)
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,4) = A142459 *)
T[n_, k_]:= T[n, k]= Sum[ (-1)^(k-j-1)*t[2*n,j+1,4], {j,0,k-1}];
Table[T[n, k], {n,12}, {k,2*n-1}]//Flatten (* G. C. Greubel, Mar 19 2022 *)

@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 A142459(n, k): return T(n, k, 4)
def A225415(n,k): return sum( (-1)^(k-j-1)*A142459(2*n, j+1) for j in (0..k-1) )
flatten([[A225415(n, k) for k in (1..2*n-1)] for n in (1..12)]) # G. C. Greubel, Mar 19 2022

T(n, k) = Sum_{j=0..k-1} (-1)^(k-j-1)*A142459(2*n, j+1). - G. C. Greubel, Mar 19 2022

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

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A002671 a(n) = 4^n*(2*n+1)!.

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

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

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A225415 Triangle read by rows: absolute values of odd-numbered rows of A225434.

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A002671 a(n) = 4^n(2n+1)!.