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 1-5 of 5 results.

A084938 Triangle read by rows: T(n,k) = Sum_{j>=0} j!*T(n-j-1, k-1) for n >= 0, k >= 0.

Original entry on oeis.org

1, 0, 1, 0, 1, 1, 0, 2, 2, 1, 0, 6, 5, 3, 1, 0, 24, 16, 9, 4, 1, 0, 120, 64, 31, 14, 5, 1, 0, 720, 312, 126, 52, 20, 6, 1, 0, 5040, 1812, 606, 217, 80, 27, 7, 1, 0, 40320, 12288, 3428, 1040, 345, 116, 35, 8, 1, 0, 362880, 95616, 22572, 5768, 1661, 519, 161, 44, 9, 1
Offset: 0

Views

Author

Philippe Deléham, Jul 16 2003; corrections Dec 17 2008, Dec 20 2008, Feb 05 2009

Keywords

Comments

Triangle T(n,k) is [0,1,1,2,2,3,3,4,4,...] DELTA [1,0,0,0,0,0,...] = A110654 DELTA A000007.
In general, the triangle [r_0,r_1,r_2,r_3,...] DELTA [s_0,s_1,s_2,s_3,...] has generating function 1/(1-(r_0*x+s_0*x*y)/(1-(r_1*x+s_1*x*y)/(1-(r_2*x+s_2*x*y)/(1-(r_3*x+s_3*x*y)/(1-...(continued fraction). See also the Formula section below.
T(n,k) = number of permutations on [n] that (i) contain a 132 pattern only as part of a 4132 pattern and (ii) start with n+1-k. For example, for n >= 1, T(n,1) = (n-1)! counts all (n-1)! permutations on [n] that start with n: either they avoid 132 altogether or the initial entry serves as the "4" in a 4132 pattern and T(4,3) = 3 counts 2134, 2314, 2341. - David Callan, Jul 20 2005
T(n,k) is the number of permutations on [n] that (i) contain a (scattered) 342 pattern only as part of a 1342 pattern and (ii) contain 1 in position k. For example, T(4,3) counts 3214, 4213, 4312. (It does not count, say, 2314 because 231 forms an offending 342 pattern.) - David Callan, Jul 20 2005
Riordan array (1,x*g(x)) where g(x) is the g.f. of the factorials (n!). - Paul Barry, Sep 25 2008
Modulo 2, this sequence becomes A106344.
T(n,k) is the number of permutations of {1,2,...,n} having k cycles such that the elements of each cycle of the permutation form an interval. - Ran Pan, Nov 11 2016
The convolution triangle of the factorial numbers. - Peter Luschny, Oct 09 2022

Examples

			From _Paul Barry_, Sep 25 2008: (Start)
Triangle [0,1,1,2,2,3,3,4,4,5,5,...] DELTA [1,0,0,0,0,...] begins
  1;
  0,      1;
  0,      1,     1;
  0,      2,     2,     1;
  0,      6,     5,     3,    1;
  0,     24,    16,     9,    4,    1;
  0,    120,    64,    31,   14,    5,   1;
  0,    720,   312,   126,   52,   20,   6,   1;
  0,   5040,  1812,   606,  217,   80,  27,   7,  1;
  0,  40320, 12288,  3428, 1040,  345, 116,  35,  8, 1;
  0, 362880, 95616, 22572, 5768, 1661, 519, 161, 44, 9, 1. (End)
From _Paul Barry_, May 14 2009: (Start)
The production matrix is
  0,   1;
  0,   1,  1;
  0,   1,  1, 1;
  0,   2,  1, 1, 1;
  0,   7,  2, 1, 1, 1;
  0,  34,  7, 2, 1, 1, 1;
  0, 206, 34, 7, 2, 1, 1, 1;
which is based on A075834. (End)

Links

Crossrefs

Cf. A003149, A051295 (row sums), A052186, A090238,
Cf. A287899.
Columns: A000007, A000142, A003149, A090595, A090319.
Diagonals: A000012, A001477, A000096, A092286, A090386, A090391, A090392, A090393, A090394.

Programs

  • Magma
    function T(n,k) // T = A084938
  if k lt 0 or k gt n then return 0;
  elif n eq 0 or k eq n then return 1;
  elif k eq 0 then return 0;
  else return (&+[Factorial(j)*T(n-j-1,k-1): j in [0..n-1]]);
  end if; return T;
end function;
[T(n,k): k in [0..n], n in [0..12]]; // G. C. Greubel, Nov 10 2022
  • Maple
    DELTA := proc(r,s,n) local T,x,y,q,P,i,j,k,t1; T := array(0..n,0..n);
for i from 0 to n do q[i] := x*r[i+1]+y*s[i+1]; od: for k from 0 to n do P[0,k] := 1; od: for i from 0 to n do P[i,-1] := 0; od:
for i from 1 to n do for k from 0 to n do P[i,k] := sort(expand(P[i,k-1] + q[k]*P[i-1,k+1])); od: od:
for i from 0 to n do t1 := P[i,0]; for j from 0 to i do T[i,j] := coeff(coeff(t1,x,i-j),y,j); od: lprint( seq(T[i,j],j=0..i) ); od: end;
# To produce the current triangle: s3 := n->floor((n+1)/2); s4 := n->if n = 0 then 1 else 0; fi; r := [seq(s3(i),i= 0..40)]; s := [seq(s4(i),i=0..40)]; DELTA(r,s,20);
# Uses function PMatrix from A357368.
PMatrix(10, n -> factorial(n - 1)); # Peter Luschny, Oct 09 2022
  • Mathematica
    a[0, 0] = 1; a[n_, k_] := a[n, k] = Sum[j! a[n - j - 1, k - 1], {j, 0, n - 1}]; Flatten[Table[a[i, j], {i, 0, 10}, {j, 0, i}]] (* T. D. Noe, Feb 22 2012 *)
DELTA[r_, s_, m_] := Module[{p, q, t, x, y}, q[k_] := x*r[[k+1]] + y*s[[k+1]]; p[0, ] = 1; p[, -1] = 0; p[n_ /; n >= 1, k_ /; k >= 0] := p[n, k] = p[n, k-1] + q[k]*p[n-1, k+1] // Expand; t[n_, k_] := Coefficient[p[n, 0], x^(n-k)*y^k]; t[0, 0] = p[0, 0]; Table[t[n, k], {n, 0, m}, {k, 0, n}]]; DELTA[Floor[Range[10]/2], Prepend[Table[0, {10}], 1], 10] (* Jean-François Alcover, Sep 12 2013, after Philippe Deléham *)
  • Sage
    def delehamdelta(R, S) :
    L = min(len(R), len(S)) + 1
    ring = PolynomialRing(ZZ, 'x')
    x = ring.gen()
    A = [Rk + x*Sk for Rk, Sk in zip(R, S)]
    C = [ring(0)] + [ring(1) for i in range(L)]
    for k in (1..L) :
        for n in range(k-1,0,-1) :
            C[n] = C[n-1] + C[n+1]*A[n-1]
        yield list(C[1])
def A084938_triangle(n) :
    for row in delehamdelta([(i+1)//2 for i in (0..n)], [0^i for i in (0..n)]):
        print(row)
A084938_triangle(10) # Peter Luschny, Jan 28 2012

Formula

The operator DELTA takes two sequences r = (r_0, r_1, ...), s = (s_0, s_1, ...) and produces a triangle T(n, k), 0 <= k <= n, as follows:
Let q(k) = x*r_k + y*s_k for k >= 0; let P(n, k) (n >= 0, k >= -1) be defined recursively by P(0, k) = 1 for k >= 0; P(n, -1) = 0 for n >= 1; P(n, k) = P(n, k-1) + q(k)*P(n-1, k+1) for n >= 1, k >= 0. Then P(n, k) is a homogeneous polynomial in x and y of degree n and T(n, k) = coefficient of x^(n-k)*y^k in P(n, 0).
T(n, n) = 1.
T(k+1, k) = A001477(k).
T(k+2, k) = A000096(k).
T(n+1, 1) = A000142(n).
T(n+2, 2) = A003149(n).
T(n+3, 3) = A090595(n).
T(n+4, 4) = A090319(n).
T(m+n, m) = Sum_{k=0..n} A090238(n, k)*binomial(m, k).
G.f. for column k: Sum_{n>=0} T(k+n, k)*x^n = (Sum_{n>=0} n!*x^n )^k.
For k>0, T(n+k, k) = Sum_{a_1 + a_2 + .. + a_k = n} (a_1)!*(a_2)!*..*(a_k)!; a_i>=0, n>=0.
T(n,k) = Sum_{j>=0} A075834(j)*T(n-1,k+j-1).
T(2n,n) = A287899(n). - Alois P. Heinz, Jun 02 2017
From G. C. Greubel, Nov 10 2022: (Start)
Sum_{k=0..n} T(n, k) = A051295(n).
Sum_{k=0..n} (-1)^k*T(n, k) = [n=0] - A052186(n-1)*[n>0]. (End)

Extensions

Name edited by Derek Orr, May 01 2015

A096161 Row sums for triangle A096162.

Original entry on oeis.org

1, 3, 8, 30, 133, 768, 5221, 41302, 369170, 3677058, 40338310, 483134179, 6271796072, 87709287104, 1314511438945, 21017751750506, 357102350816602, 6424883282375340, 122025874117476166, 2439726373093186274
Offset: 1

Views

Author

Alford Arnold, Jun 18 2004

Keywords

Comments

Also, partitions such that a set of k equal terms are labeled 1 through k and can appear in any order. For example, the partition 3+2+2+2+1+1+1+1 of 13 appears 1!*3!*4!=144 times because there are 1! ways to order the one "3," 3! ways to order the three "2"s, ... - Christian G. Bower, Jan 17 2006

Examples

			1 1 2 1 3 6 1 4 6 12 24 ... A036038
1 1 1 1 3 1 1 4 3 6 1 ... A036040
1 1 2 1 1 6 1 1 2 2 24 ... A096162
so a(n) begins 1 3 8 30 ... A096161

Links

Crossrefs

Cf. A005651, A000110, A036038, A036040, A096162, A110143, A287899.

Programs

  • Mathematica
    nmax = 25; Rest[CoefficientList[Series[Product[Sum[k!*x^(j*k), {k, 0, nmax/j}], {j, 1, nmax}], {x, 0, nmax}], x]] (* Vaclav Kotesovec, Aug 10 2019 *)
m = 25; Rest[CoefficientList[Series[Product[-Gamma[0, -1/x^j] * Exp[-1/x^j], {j, 1, m}] / x^(m*(m + 1)/2), {x, 0, m}], x]] (* Vaclav Kotesovec, Dec 07 2020 *)
  • PARI
    { my(n=25); Vec(prod(k=1, n, O(x*x^n) + sum(r=0, n\k, x^(r*k)*r!))) }

Formula

G.f.: B(x)*B(x^2)*B(x^3)*... where B(x) is g.f. of A000142. - Christian G. Bower, Jan 17 2006
G.f.: Product_{k>0} Sum_{r>=0} x^(r*k)*r!. - Andrew Howroyd, Dec 22 2017
a(n) ~ n! * (1 + 1/n^2 + 2/n^3 + 7/n^4 + 28/n^5 + 121/n^6 + 587/n^7 + 3205/n^8 + 19201/n^9 + 123684/n^10 + ...), for coefficients see A293266. - Vaclav Kotesovec, Aug 10 2019

Extensions

More terms from Vladeta Jovovic, Jun 22 2004

A309652 a(n) = [x^n] B(x)^n, where B(x) is g.f. of A000312.

Original entry on oeis.org

1, 1, 9, 106, 1493, 24276, 448122, 9301251, 215547845, 5541171496, 156997349684, 4870353700532, 164366482285898, 5998207807965543, 235388194276592723, 9884482616014596546, 442206843338189113445, 20995082225203329126384, 1054247070579064423466016
Offset: 0

Views

Author

Vaclav Kotesovec, Aug 11 2019

Keywords

Links

Crossrefs

Cf. A287899, A326985, A326986.

Programs

  • Maple
    B:= proc(n) option remember; n^n end:
b:= proc(n, i) option remember; `if`(n=0, 1, `if`(i=1, B(n),
      (h-> add(b(j, h)*b(n-j, i-h), j=0..n))(iquo(i, 2))))
    end:
a:= n-> b(n$2):
seq(a(n), n=0..20);  # Alois P. Heinz, Aug 23 2019
  • Mathematica
    Table[SeriesCoefficient[(1+Sum[k^k*x^k, {k, 1, n}])^n, {x, 0, n}], {n, 0, 20}]

Formula

a(n) ~ exp(exp(-1)) * n^(n+1).

A293471 a(n) = [x^n] (1/(1 - 2*x/(1 - 2*x/(1 - 4*x/(1 - 4*x/(1 - 6*x/(1 - 6*x/(1 - ...))))))))^n, a continued fraction.

Original entry on oeis.org

1, 2, 20, 248, 3472, 53152, 878144, 15577984, 296411392, 6054973952, 132994708480, 3144712222720, 80063883022336, 2192452931723264, 64427309553434624, 2025284853319303168, 67859418068644069376, 2414526405567056052224, 90909088845844899430400, 3610058425696043667030016
Offset: 0

Views

Author

Ilya Gutkovskiy, Oct 09 2017

Keywords

Links

Crossrefs

Cf. A000165, A052712, A126674, A287899, A293470.

Programs

  • Mathematica
    Table[SeriesCoefficient[1/(1 + ContinuedFractionK[-2 Floor[(k + 1)/2] x, 1, {k, 1, n}])^n, {x, 0, n}], {n, 0, 19}]
Table[SeriesCoefficient[Sum[(2 k)!! x^k, {k, 0, n}]^n, {x, 0, n}], {n, 0, 19}]

Formula

a(n) ~ sqrt(Pi) * 2^(n + 1/2) * n^(n + 3/2) / exp(n-1). - Vaclav Kotesovec, Sep 16 2021

A293470 a(n) = [x^n] (1/(1 - x/(1 - 2*x/(1 - 3*x/(1 - 4*x/(1 - 5*x/(1 - 6*x/(1 - ...))))))))^n, a continued fraction.

Original entry on oeis.org

1, 1, 7, 64, 691, 8506, 117586, 1811902, 30977059, 585159526, 12157511122, 276365651992, 6835179127294, 182885413524568, 5265255383238592, 162296482607602714, 5332203008816278819, 185989603728568482598, 6863252473075010369626, 267102762222709967674384, 10932746393513621360731066
Offset: 0

Views

Author

Ilya Gutkovskiy, Oct 09 2017

Keywords

Links

Crossrefs

Cf. A001147, A034430, A113551, A287899, A293471.

Programs

  • Mathematica
    Table[SeriesCoefficient[1/(1 + ContinuedFractionK[-k x, 1, {k, 1, n}])^n, {x, 0, n}], {n, 0, 20}]
Table[SeriesCoefficient[Sum[(2 k - 1)!! x^k, {k, 0, n}]^n, {x, 0, n}], {n, 0, 20}]

Formula

a(n) ~ 2^(n + 1/2) * n^(n+1) / exp(n - 1/2). - Vaclav Kotesovec, Sep 16 2021
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