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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A227343 Matrix inverse of triangle A227342.

Original entry on oeis.org

1, 1, 1, 3, 3, 1, 13, 13, 6, 1, 75, 75, 37, 10, 1, 541, 541, 270, 85, 15, 1, 4683, 4683, 2341, 770, 170, 21, 1, 47293, 47293, 23646, 7861, 1890, 308, 28, 1, 545835, 545835, 272917, 90930, 22491, 4158, 518, 36, 1, 7087261, 7087261, 3543630, 1181125, 294525, 57351, 8400, 822, 45, 1
Offset: 0

Views

Author

Peter Bala, Jul 11 2013

Keywords

Comments

The e.g.f. has the form A(t)*exp(x*B(t)), where A(t) = 1/(2 - exp(t)) and B(t) = exp(t) - 1. Thus the row polynomials of this triangle form a Sheffer sequence for the pair (1 - t, log(1 + t)) (see Roman, p.17).
Let x_(k) := x*(x-1)*...*(x-k+1) denote the k-th falling factorial polynomial. Define a sequence x_[n] of basis polynomials for the polynomial algebra C[x] by setting x_[0] = 1, and setting x_[n] = x_(n-1)*(x - 2*n + 1) for n >= 1. The sequence begins [1, x-1, x*(x-3), x*(x-1)*(x-5), x*(x-1)*(x-2)*(x-7), ...]. Then this is the triangle of connection constants for expressing the monomial polynomials x^n as a linear combination of the basis x_[k], that is, x^n = sum {k = 0..n} T(n,k)*x_[k]. An example is given below.

Examples

			Triangle begins
n\k|   0    1    2    3    4    5
= = = = = = = = = = = = = = = = =
0 |   1
1 |   1    1
2 |   3    3    1
3 |  13   13    6    1
4 |  75   75   37   10    1
5 | 541  541  270   85   15    1
...
Connection constants. Row 4 = [75,75,37,10,1]: Thus
75 + 75*(x - 1) + 37*x*(x - 3) + 10*x*(x - 1)*(x - 5)+ x*(x - 1)*(x - 2)*(x - 7) = x^4.

References

  • S. Roman, The umbral calculus, Pure and Applied Mathematics 111, Academic Press Inc., New York, 1984. Reprinted by Dover in 2005.

Links

Crossrefs

A000670 (columns 1 and 2), A048993, A059099 (row sums), A105794, A227342 (matrix inverse).

Programs

  • Mathematica
    T[n_, k_] := n!/k! SeriesCoefficient[Series[1/(2 - Exp[t]) (Exp[t] - 1)^k, {t, 0, n}], n]
Flatten[Table[T[n, k], {n, 0, 12}, {k, 0, n}]]
U[n_, k_] := n!/k! SeriesCoefficient[Series[1/(1 - t^2) (t/Log[1 + t])^(n + 1), {t, 0, n - k}], n - k]
Flatten[Table[U[n, k], {n, 0, 8}, {k, 0, n}]] (* Emanuele Munarini, Dec 21 2016 *)

Formula

E.g.f.: 1/(2 - exp(t))*exp(x*(exp(t) - 1)) = 1 + (1 + x)*t + (3 + 3*x + x^2)*t^2/2! + (13 + 13*x + 6*x^2 + x^3)*t^3/3! + ....
Recurrence equation: T(n,0) = A000670(n), and for k >= 1, T(n,k) = 1/k*sum {i = 1..n} binomial(n,i)*T(n-i,k-1).
The row polynomials R(n,x) satisfy the Sheffer identity R(n,x + y) = sum {k = 0..n} binomial(n,k)*Bell(k,y)*R(n-k,x), where Bell(k,y) is the Bell or exponential polynomial (row polynomials of A048993).
The row polynomials also satisfy d/dx(R(n,x)) = sum {k = 0..n-1} binomial(n,k)*R(k,x).
Row sums A059099. Column 1 and column 2 = A000670. 1 + 2*column 3 = A000670 (apart from the first two terms).
From Emanuele Munarini, Dec 21 2016: (Start)
T(n,k) = (n!/k!)*[t^n](exp(t)-1)^k/(2-exp(t)).
T(n,k) = (n!/k!)*[t^(n-k)](t/log(1+t))^(n+1)/(1-t^2). (End)

A241765 a(n) = n*(n + 1)*(n + 2)*(3*n + 17)/24.

Original entry on oeis.org

0, 5, 23, 65, 145, 280, 490, 798, 1230, 1815, 2585, 3575, 4823, 6370, 8260, 10540, 13260, 16473, 20235, 24605, 29645, 35420, 41998, 49450, 57850, 67275, 77805, 89523, 102515, 116870, 132680, 150040, 169048, 189805, 212415, 236985, 263625, 292448
Offset: 0

Views

Author

Bruno Berselli, Apr 28 2014

Keywords

Comments

Equivalently, Sum_{i=0..n} (i+4)*A000217(i).
Sequences of the type Sum_{i=0..n} (i+k)*A000217(i):
k = 0, A001296: 0, 1, 7, 25, 65, 140, 266, 462, ...
k = 1, A000914: 0, 2, 11, 35, 85, 175, 322, 546, ...
k = 2, A050534: 0, 3, 15, 45, 105, 210, 378, 630, ... (deleting two 0)
k = 3, A215862: 0, 4, 19, 55, 125, 245, 434, 714, ...
k = 4, a(n): 0, 5, 23, 65, 145, 280, 490, 798, ...
k = 5, A239568: 0, 6, 27, 75, 165, 315, 546, 882, ...
Antidiagonal sums (without 0) give A034263: 1, 9, 39, 119, 294, ...
Diagonal: 1, 11, 45, 125, 280, 546, ... is A051740.
Also: k = -1 gives A050534 deleting a 0; k = -2 gives 0 followed by A059302.
After 0, partial sums of A212343 and third column of A118788.
This sequence is even related to A005286 by a(n) = n*A005286(n) - Sum_{i=0..n-1} A005286(i).

Examples

			a(7) = 4*0 + 5*1 + 6*3 + 7*6 + 8*10 + 9*15 + 10*21 + 11*28 = 798.

Links

Crossrefs

Cf. A000217, A005286, A118788, A212343, A227342.
Cf. similar sequences A000914, A001296, A050534, A059302, A215862, A239568 (see table in Comments lines).

Programs

  • Magma
    /* By first comment: */ k:=4; A000217:=func; [&+[(i+k)*A000217(i): i in [0..n]]: n in [0..40]];
  • Maple
    A241765:=n->n*(n + 1)*(n + 2)*(3*n + 17)/24; seq(A241765(n), n=0..40); # Wesley Ivan Hurt, May 09 2014
  • Mathematica
    Table[n (n + 1) (n + 2) (3 n + 17)/24, {n, 0, 40}] (* or *) LinearRecurrence[{5, -10, 10, -5, 1}, {0, 5, 23, 65, 145}, 40]
CoefficientList[Series[x (5 - 2 x)/(1 - x)^5, {x, 0, 40}], x] (* Vincenzo Librandi, May 09 2014 *)
  • Maxima
    makelist(coeff(taylor(x*(5-2*x)/(1-x)^5, x, 0, n), x, n), n, 0, 40);
  • PARI
    a(n)=n*(n+1)*(n+2)*(3*n+17)/24 \\ Charles R Greathouse IV, Oct 07 2015
  • PARI
    x='x+O('x^99); concat(0, Vec(x*(5-2*x)/(1-x)^5)) \\ Altug Alkan, Apr 10 2016
  • Sage
    [n*(n+1)*(n+2)*(3*n+17)/24 for n in (0..40)]

Formula

G.f.: x*(5 - 2*x)/(1 - x)^5.
a(n) = 5*a(n-1) - 10*a(n-2) + 10*a(n-3) - 5*a(n-4) + a(n-5).
a(n) = A227342(A055998(n+1)).
a(n) = Sum_{j=0..n+2} (-1)^(n-j)*binomial(-j,-n-2)*S1(j,n), S1 Stirling cycle numbers A132393. - Peter Luschny, Apr 10 2016
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