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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A187543 Binomial convolutions of the central Lah numbers (A187535).

Original entry on oeis.org

1, 4, 80, 2832, 144576, 9660480, 798468480, 78670609920, 9002061573120, 1173384611804160, 171641216823552000, 27843893955582566400, 4961007038613633638400, 963075987422089673932800, 202333751987206944654950400
Offset: 0

Views

Author

Emanuele Munarini, Mar 11 2011

Keywords

Links

Crossrefs

Cf. A187536, A008297, A111596, A187536, A187538, A187539, A187540, A187542, A187544, A187545, A187546, A187547, A187548.

Programs

  • Maple
    a := n -> if n=0 then 1 else binomial(2*n-1,n-1)*(2*n)!/n! fi;
seq(add(binomial(n,k)*a(k)*a(n-k), k=0..n),n=0..12);
  • Mathematica
    a[n_] := If[n == 0, 1, Binomial[2n - 1, n - 1](2n)!/n!]
Table[Sum[Binomial[n, k]a[k]a[n - k], {k, 0, n}], {n, 0, 20}]
CoefficientList[Series[(1/2 + EllipticK[16*x]/Pi)^2, {x, 0, 20}], x] * Range[0, 20]! (* Vaclav Kotesovec, Oct 06 2019 *)
  • Maxima
    a(n):= if n=0 then 1 else binomial(2*n-1,n-1)*(2*n)!/n!;
makelist(sum(binomial(n,k)*a(k)*a(n-k),k,0,n),n,0,12);

Formula

a(n) = sum(binomial(n,k)*L(k)*L(n-k),k=0..n), where L(n) is a central Lah number.
E.g.f.: (1/2 + 1/Pi*K(16x))^2, where K(z) is the elliptic integral of the first kind (defined as in Mathematica).
Recurrence: (n-1)*n^2*(4*n^2-15*n+13)*a(n) = 4*(n-1)*(48*n^5-292*n^4+672*n^3-747*n^2+399*n-76)*a(n-1) - 32*(96*n^7-1000*n^6+4408*n^5-10628*n^4+15034*n^3-12312*n^2+5265*n-854)*a(n-2) + 1024*(2*n-5)^2*(4*n^2-7*n+2)*(n-2)^4*a(n-3). - Vaclav Kotesovec, Aug 10 2013
a(n) ~ n! * log(n) * 2^(4*n-1) / (Pi^2 * n) * (1 + (gamma + Pi + 4*log(2)) / log(n)), where gamma is the Euler-Mascheroni constant A001620. - Vaclav Kotesovec, Oct 06 2019

A187544 Stirling transform (of the second kind) of the central Lah numbers (A187535).

Original entry on oeis.org

1, 2, 38, 1310, 66254, 4428782, 368444078, 36691056110, 4256199137774, 563672814445742, 83921091641375918, 13875375391723852910, 2522552600160248918894, 500141581330626431059502, 107400097037199576065830958
Offset: 0

Views

Author

Emanuele Munarini, Mar 11 2011

Keywords

Links

Crossrefs

Cf. A187536, A008297, A111596, A187536, A187538, A187539, A187540, A187542, A187543, A187545, A187546, A187547, A187548.

Programs

  • Maple
    a := n -> if n=0 then 1 else binomial(2*n-1,n-1)*(2*n)!/n! fi;
seq(sum(combinat[stirling2](n,k)*a(k), k=0..n),n=0..12);
  • Mathematica
    a[n_] := If[n == 0, 1, Binomial[2n - 1, n - 1](2n)!/n!]
Table[Sum[StirlingS2[n, k]a[k], {k, 0, n}], {n, 0, 20}]
CoefficientList[Series[1/2 + EllipticK[16*(E^x - 1)]/Pi, {x, 0, 20}], x] * Range[0, 20]! (* Vaclav Kotesovec, Oct 06 2019 *)
  • Maxima
    a(n):= if n=0 then 1 else binomial(2*n-1,n-1)*(2*n)!/n!;
makelist(sum(stirling2(n,k)*a(k),k,0,n),n,0,12);

Formula

a(n) = sum(S(n,k)*L(k),k=0..n), where S(n,k) are the Stirling numbers of the second kind and L(n) are the central Lah numbers.
E.g.f.: 1/2 + 1/Pi*K(16(exp(x)-1)) where K(z) is the elliptic integral of the first kind (defined as in Mathematica).
a(n) ~ n! / (2*Pi*n * (log(17/16))^n). - Vaclav Kotesovec, Oct 06 2019

A187545 Stirling transform (of the first kind) of the central Lah numbers (A187535).

Original entry on oeis.org

1, 2, 38, 1312, 66408, 4442088, 369791064, 36848702784, 4277191653888, 566809715422464, 84441103242634176, 13970100487593468480, 2541362625439551554880, 504185908064687887996800, 108336183242510523080868480
Offset: 0

Views

Author

Emanuele Munarini, Mar 11 2011

Keywords

Links

Crossrefs

Cf. A187536, A008297, A111596, A187536, A187538, A187539, A187540, A187542, A187543, A187544, A187546, A187547, A187548.

Programs

  • Maple
    lahc := n -> if n=0 then 1 else binomial(2*n-1,n-1)*(2*n)!/n! fi;
seq(add(abs(combinat[stirling1](n,k))*lahc(k), k=0..n), n=0..20);
  • Mathematica
    lahc[n_] := If[n == 0, 1, Binomial[2n - 1, n - 1](2n)!/n!]
Table[Sum[Abs[StirlingS1[n, k]]*lahc[k], {k, 0, n}], {n, 0, 20}]
  • Maxima
    lahc(n):= if n=0 then 1 else binomial(2*n-1,n-1)*(2*n)!/n!;
makelist(sum(abs(stirling1(n,k))*lahc(k),k,0,n),n,0,12);

Formula

a(n) = sum(s(n,k)*L(k), k=0..n), where s(n,k) are the (signless) Stirling numbers of the first kind and L(n) are the central Lah numbers.
E.g.f.: 1/2 + 1/Pi*K(-16*log(1-x)), where K(z) is the elliptic integral of the first kind (defined as in Mathematica).
a(n) ~ n! / (2*Pi*n * (1 - exp(-1/16))^n). - Vaclav Kotesovec, Apr 10 2018

A187546 Stirling transform (of the first kind, with signs) of the central Lah numbers (A187535).

Original entry on oeis.org

1, 2, 34, 1096, 51984, 3262488, 254943384, 23853046656, 2600024557248, 323588157732096, 45276442446814656, 7035574740347812800, 1202158966644148296000, 224022356544364922931840, 45215509996613004825121920
Offset: 0

Views

Author

Emanuele Munarini, Mar 11 2011

Keywords

Links

Crossrefs

Cf. A187536, A008297, A111596, A187536, A187538, A187539, A187540, A187542, A187543, A187544, A187545, A187547, A187548.

Programs

  • Maple
    lahc := n -> if n=0 then 1 else binomial(2*n-1,n-1)*(2*n)!/n! fi;
seq(add(combinat[stirling1](n,k)*lahc(k), k=0..n), n=0..20);
  • Mathematica
    lahc[n_] := If[n == 0, 1, Binomial[2n - 1, n - 1](2n)!/n!]
Table[Sum[StirlingS1[n, k]*lahc[k], {k, 0, n}], {n, 0, 20}]
  • Maxima
    lahc(n):= if n=0 then 1 else binomial(2*n-1,n-1)*(2*n)!/n!;
makelist(sum(stirling1(n,k)*lahc(k),k,0,n),n,0,12);

Formula

a(n) = sum((-1)^(n-k)*s(n,k)*L(k), k=0..n), where s(n,k) are the (signless) Stirling numbers of the first kind and L(n) are the central Lah numbers.
E.g.f.: 1/2 + 1/Pi*K(16*log(1+x)), where K(z) is the elliptic integral of the first kind (defined as in Mathematica).
a(n) ~ n! / (2*Pi*n * (exp(1/16) - 1)^n). - Vaclav Kotesovec, Apr 10 2018

A187547 L(n)H(n+1), product of the central Lah number L(n) and the harmonic number H(n).

Original entry on oeis.org

1, 3, 66, 2500, 134260, 9335088, 796938912, 80671795776, 9446603680800, 1256254443100800, 187033518310129920, 30821040496874234880, 5569495264653352381440, 1095113648992295923200000, 232773183612995427763200000, 53186532693832607435089920000
Offset: 0

Views

Author

Emanuele Munarini, Mar 11 2011

Keywords

Crossrefs

Cf. A187536, A008297, A111596, A187536, A187538, A187539, A187540, A187542, A187543, A187544, A187545, A187546, A187548.

Programs

  • Maple
    a := n -> if n=0 then 1 else binomial(2*n-1,n-1)*(2*n)!/n! fi;
seq(a(n)*sum(1/k,k=1..n+1),n=0..12);
  • Mathematica
    a[n_] := If[n == 0, 1, Binomial[2n - 1, n - 1](2n)!/n!]
Table[a[n]HarmonicNumber[n + 1], {n, 0, 20}]
  • Maxima
    a(n):= if n=0 then 1 else binomial(2*n-1,n-1)*(2*n)!/n!;
makelist(a(n)*sum(1/k,k,1,n+1),n,0,12);

Formula

Recurrence:
(n+3)(n+2)(n+1)a(n+2)-4(2n+3)^2(2n+5)(n+1)a(n+1)+16(2n+3)^2(2n+1)^2(n+2)a(n)-144delta(n,0)=0.

A136656 Coefficients for rewriting generalized falling factorials into ordinary falling factorials.

Original entry on oeis.org

1, 0, -2, 0, 6, 4, 0, -24, -36, -8, 0, 120, 300, 144, 16, 0, -720, -2640, -2040, -480, -32, 0, 5040, 25200, 27720, 10320, 1440, 64, 0, -40320, -262080, -383040, -199920, -43680, -4032, -128, 0, 362880, 2963520, 5503680, 3764880, 1142400, 163968, 10752, 256, 0, -3628800, -36288000, -82978560
Offset: 0

Views

Author

Wolfdieter Lang, Feb 22 2008, Sep 09 2008

Keywords

Comments

Generalization of (signed) Lah number triangle A008297 (amended with a trivial row n=0 and a column k=0 in order to have a Sheffer triangle structure of the Jabotinsky type).
product(s*t-j,j=0..n-1) := fallfac(s*t,n) (falling factorial with n factors) is called generalized factorial of t of order n and scale parameter s in the Charalambides reference p. 301 ch. 8.4.
The s-family of triangles L(s;n,k) (in the Charalambides reference called C(n,k;-s)) is defined for integer s by fallfac(-s*t,n) = ((-1)^n)*risefac(s*t,n) = sum(L(s;n,k)*fallfac(t,k),k=0..n), n>=0. risefac(x,n):=product(x+j,j=0..n-1) for the rising factorials.
For positive s the signless triangles |L(s;n,k)| = L(s;n,k)*(-1)^n satisfies risefac(s*t,n) = sum(|L(s;n,k)|*fallfac(t,k),k=0..n), n>=0.
For negative s see the combinatorial interpretation given in the Charalambides reference, Example 8.8, p. 313: Coupon collector's problem.
|T(n,k)| = B_{n,k}((j+2)!; j>=0) where B_{n,k} are the partial Bell polynomials. - Peter Luschny, May 11 2015

Examples

			Triangle starts:
[1]
[0,   -2]
[0,    6,    4]
[0,  -24,  -36,   -8]
[0,  120,  300,  144,  16]
...
Recurrence: a(4,2) = -7*a(3,2)-2*a(3,1) = -7*(-36) -2*(-24) = 300.
a(4,2)=300 as sum over the M3 numbers A036040 for the 2 parts partitions of 4: 4*fallfac(-2,1)^1*fallfac(-2,3)^1 + 3*fallfac(-2,2)^2 = 4*(-2)*(-24)+3*6^2 = 300.
Row n=3: [0,-24,-36,-8] for the coefficients in rewriting fallfac(-2*t,3)=((-1)^3)*risefac(2*t,3) = ((-1)^3)*(2*t)*(2*t+1)*(2*t+2) = 0*1 -24*t -36*t*(t-1) -8*t*(t-1)*(t-2).

References

  • Ch. A. Charalambides, Enumerative Combinatorics, Chapman & Hall/CRC, Boca Raton, Florida, 2002, ch. 8.4 p. 301 ff. Table 8.3 (with row n=0 and column k=0 and s=-2).

Links

Crossrefs

Column sequences (unsigned) 2*A001710, 4*A136659, 8*A136660, 16*A136661 for k=1..4.
Cf. A136657 without row n=0 and column k=0, divided by 2.

Programs

  • Maple
    # The function BellMatrix is defined in A264428.
BellMatrix(n -> (-1)^(n+1)*(n+2)!, 9); # Peter Luschny, Jan 27 2016
  • Mathematica
    fallfac[n_, k_] := Pochhammer[n - k + 1, k]; a[n_, k_] := Sum[(-1)^(k - r)*Binomial[k, r]*fallfac[-2*r, n], {r, 0, k}]/k!; Table[a[n, k], {n, 0, 9}, {k, 0, n}] // Flatten (* Jean-François Alcover, Jul 09 2013 *)
BellMatrix[f_Function, len_] := With[{t = Array[f, len, 0]}, Table[BellY[n, k, t], {n, 0, len - 1}, {k, 0, len - 1}]];
rows = 10;
M = BellMatrix[(-1)^(# + 1)*(# + 2)!&, rows];
Table[M[[n, k]], {n, 1, rows}, {k, 1, n}] // Flatten (*Jean-François Alcover, Jun 23 2018, after Peter Luschny *)
  • Sage
    @CachedFunction
def T(n, k):  # unsigned case
    if k == 0: return 1 if n == 0 else 0
    return sum(T(n-j-1,k-1)*(j+1)*(j+2)*gamma(n)/gamma(n-j) for j in (0..n-k))
for n in range(7): [T(n,k) for k in (0..n)] # Peter Luschny, Mar 31 2015

Formula

Recurrence: a(n,k) = 0 if n
E.g.f. column k: (1/(1+x)^2 - 1)^k/k!, k>=0. From the Charalambides reference Theorem 8.14, p. 305 for s=-2.(hence a Sheffer triangle of Jabotinsky type).
a(n,k) = sum(((-1)^(k-r))*binomial(k,r)*fallfac(-2*r,n),r=0..k)/k!, n>=k>=0. From the Charalambides reference Theorem 8.15, p. 306 for s=-2.
a(n,k) = sum(S1(n,r)*S2(r,k)*(-2)^r,r=k..n) with the Stirling numbers S1(n,r)= A048993(n,r) and S2(r,k)= A048993(r,k). From the Charalambides reference Theorem 8.13, p.304 for s=-2.
a(n,k) = sum(M_3(n,k,q)*product(fallfac(-2,j)^e(n,m,q,j),j=1..n),q=1..p(n,k)) if n>=k>=1, else 0. Here p(n,k)=A008284(n,m), the number of k parts partitions of n, the M_3 partition numbers are given in A036040 and e(n,m,q,j) is the exponent of j in the q-th k parts partition of n. Rewritten eq. (8.50), Theorem 8.16, p. 307, from the Charalambides reference for s=-2.
Recurrence for the unsigned case: a(n,k) = Sum_{j=0..n-k} a(n-j-1,k-1)*C(n-1,j)*(j+2)! if k<>0 else k^n. - Peter Luschny, Mar 31 2015

A185422 Forests of k increasing plane unary-binary trees on n nodes. Generalized Stirling numbers of the second kind associated with A185415.

Original entry on oeis.org

1, 1, 1, 3, 3, 1, 9, 15, 6, 1, 39, 75, 45, 10, 1, 189, 459, 330, 105, 15, 1, 1107, 3087, 2709, 1050, 210, 21, 1, 7281, 23535, 23814, 11109, 2730, 378, 28, 1, 54351, 197235, 228285, 122850, 36099, 6174, 630, 36, 1
Offset: 1

Views

Author

Peter Bala, Jan 28 2011

Keywords

Comments

An increasing tree is a labeled rooted tree with the property that the sequence of labels along any path starting from the root is increasing.
A080635 enumerates increasing plane (ordered) unary-binary trees with n nodes labeled from the set {1,2,...n}. The entry T(n,k) of the present table counts forests of k increasing plane unary-binary trees having n nodes in total. See below for an example.
The Stirling number of the second kind Stirling2(n,k) counts the partitions of the set [n] set into k blocks. Arranging the elements in each block in ascending numerical order provides an alternative combinatorial interpretation for Stirling2(n,k) as counting forests of k increasing unary trees on n nodes. Thus we may view the present array as generalized Stirling numbers of the second kind (associated with A080635 or with the polynomials P(n,x) of A185415 - see formulas (1) and (2) below).
For a table of ordered forests of increasing plane unary-binary trees see A185423. For the enumeration of forests and ordered forests in the non-plane case see A147315 and A185421.
The Bell transform of A080635(n+1). For the definition of the Bell transform see A264428. - Peter Luschny, Jan 18 2016

Examples

			Triangle begins
n\k|....1......2......3......4......5......6......7
===================================================
..1|....1
..2|....1......1
..3|....3......3......1
..4|....9.....15......6......1
..5|...39.....75.....45.....10......1
..6|..189....459....330....105.....15......1
..7|.1107...3087...2709...1050....210.....21......1
..
Examples of the recurrence:
T(5,1) = 39 = T(4,0)+1*T(4,1)+2*T(4,2) = 1*9+2*15;
T(6,3) = 330 = T(5,2)+3*T(5,3)+3*4*T(5,4) = 75+3*45+12*10.
Examples of forests:
T(4,1) = 9. The 9 plane increasing unary-binary trees on 4 nodes are shown in the example section of A080635.
T(4,2) = 15. The 15 forests consisting of two plane increasing unary-binary trees on 4 nodes consist of the 12 forests
......... ......... ...3.....
.2...3... .3...2... ...|.....
..\./.... ..\./.... ...2.....
...1...4. ...1...4. ...|.....
......... ......... ...1...4.
.
......... ......... ...4.....
.2...4... .4...2... ...|.....
..\./.... ..\./.... ...2.....
...1...3. ...1...3. ...|.....
......... ......... ...1...3.
.
......... ......... ...4.....
.3...4... .4...3... ...|.....
..\./.... ..\./.... ...3.....
...1...2. ...1...2. ...|.....
......... ......... ...1...2.
......... ......... ...4.....
.3...4... .4...3... ...|.....
..\./.... ..\./.... ...3.....
...2...1. ...2...1. ...|.....
......... ......... ...2...1.
.
and the three remaining forests
......... ......... ..........
..2..4... ..3..4... ..4...3...
..|..|... ..|..|... ..|...|...
..1..3... ..1..2... ..1...2...
......... ......... ..........

References

  • F. Bergeron, Ph. Flajolet and B. Salvy, Varieties of Increasing Trees, in Lecture Notes in Computer Science vol. 581, ed. J.-C. Raoult, Springer 1922, pp. 24-48.

Links

Crossrefs

Cf. A080635, A008277, A185415, A147315, A185423.

Programs

  • Maple
    #A185422
P := proc(n,x)
description 'polynomial sequence P(n,x) A185415'
if n = 0
return 1
else
return x*(P(n-1,x-1)-P(n-1,x)+P(n-1,x+1))
end proc:
with combinat:
T:= (n,k) -> 1/k!*add ((-1)^(k-j)*binomial(k,j)*P(n,j),j = 0..k):
for n from 1 to 10 do
seq(T(n,k),k = 1..n);
end do;
  • Mathematica
    t[n_, k_] := t[n, k] = t[n-1, k-1] + k*t[n-1, k] + k*(k+1)*t[n-1, k+1]; t[n_, n_] = 1; t[n_, k_] /; Not[1 <= k <= n] = 0; Table[t[n, k], {n, 1, 9}, {k, 1, n}] // Flatten (* Jean-François Alcover, Nov 22 2012, from given recurrence *)
  • PARI
    {T(n,k)=if(n<1||k<1||k>n,0,if(n==k,1,T(n-1,k-1)+k*T(n-1,k)+k*(k+1)*T(n-1,k+1)))}
  • PARI
    {T(n,k)=round(n!*polcoeff(polcoeff(exp(y*(-1/2+sqrt(3)/2*tan(sqrt(3)/2*x+Pi/6+x*O(x^n)))+y*O(y^k)),n,x),k,y))}
  • Sage
    # uses[bell_matrix from A264428]
# Adds a column 1,0,0,0, ... at the left side of the triangle.
bell_matrix(lambda n: A080635(n+1), 10) # Peter Luschny, Jan 18 2016

Formula

TABLE ENTRIES
(1)... T(n,k) = (1/k!)*Sum_{j=0..k} (-1)^(k-j)*binomial(k,j)*P(n,j),
where P(n,x) are the polynomials described in A185415.
Compare (1) with the formula for the Stirling numbers of the second kind
(2)... Stirling2(n,k) = 1/k!*Sum_{j=0..k} (-1)^(k-j)*binomial(k,j)*j^n.
RECURRENCE RELATION
(3)... T(n+1,k) = T(n,k-1) + k*T(n,k) + k*(k+1)*T(n,k+1).
GENERATING FUNCTION
Let E(t) = 1/2 + sqrt(3)/2*tan(sqrt(3)/2*t + Pi/6) be the e.g.f. for A080635.
The e.g.f. for the present triangle is
(4)... exp{x*(E(t)-1)} = Sum_{n>=0} R(n,x)*t^n/n!
= 1 + x*t + (x+x^2)*t^2/2! + (3*x+3*x^2+x^3)*t^3/3! + ....
ROW POLYNOMIALS
The row generating polynomials R(n,x) satisfy the recurrence
(5)... R(n+1,x) = x*{R(n,x)+R'(n,x)+R''(n,x)},
where the prime ' indicates differentiation with respect to x.
RELATIONS WITH OTHER SEQUENCES
Column 1 is A080635.
k!*T(n,k) counts ordered forests A185423(n,k).
The row polynomials R(n,x) are given by D^n(exp(x*t)) evaluated at t = 0, where D is the operator (1+t+t^2)*d/dt. Cf. A147315 and A008297. - Peter Bala, Nov 25 2011

A134151 Triangle of numbers obtained from the partition array A134150.

Original entry on oeis.org

1, 4, 1, 28, 4, 1, 280, 44, 4, 1, 3640, 392, 44, 4, 1, 58240, 5544, 456, 44, 4, 1, 1106560, 80640, 5992, 456, 44, 4, 1, 24344320, 1519840, 88256, 6248, 456, 44, 4, 1, 608608000, 31420480, 1631392, 90048, 6248, 456, 44, 4, 1, 17041024000, 766525760, 33293120
Offset: 1

Views

Author

Wolfdieter Lang Nov 13 2007

Keywords

Comments

This triangle is named S2(4)'.
In the same manner the unsigned Lah triangle A008297 is obtained from the partition array A130561.

Examples

			[1]; [4,1]; [28,4,1]; [280,44,4,1]; [3640,392,44,4,1];...

Links

Crossrefs

Cf. A134152 (row sums). A134272 (alternating row sums).
Cf. A134146 (S2(3)' triangle).

Formula

a(n,m)=sum(product(S2(4;j,1)^e(n,m,q,j),j=1..n),q=1..p(n,m)) if n>=m>=1, else 0. Here p(n,m)=A008284(n,m), the number of m parts partitions of n and e(n,m,q,j) is the exponent of j in the q-th m part partition of n. S2(4;j,1)= A007559(j) = A035469(j,1) = (3*j-2)!!!.

A248045 (2*(n-1))! * (2*n-1)! / (n * (n-1)!^3).

Original entry on oeis.org

1, 6, 120, 4200, 211680, 13970880, 1141620480, 111307996800, 12614906304000, 1629845894476800, 236475822507724800, 38072607423743692800, 6735922851893114880000, 1299070835722243584000000, 271245990498804460339200000, 60962536364606302461235200000
Offset: 1

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Author

Reinhard Zumkeller, Sep 30 2014

Keywords

Comments

Central terms in triangles of Lah numbers: a(n) = - A008297(2*n-1,n) = A105278(2*n-1,n) = A000891(n-1)*A000142(n) = A000894(n-1)*A000142(n-1).
a(n) = n * A204515(n-1). - Reinhard Zumkeller, Oct 19 2014

Links

Crossrefs

Cf. A000142, A000891, A000894, A008297, A105278, A204515.
Cf. A187535 (Central Lah numbers).

Programs

  • Haskell
    a248045 n = a000891 (n - 1) * a000142 n

Formula

n*a(n) = 4*(2*n-1)*(2*n-3)*a(n-1). - R. J. Mathar, Oct 07 2014

A001778 Lah numbers: a(n) = n!*binomial(n-1,5)/6!.

Original entry on oeis.org

1, 42, 1176, 28224, 635040, 13970880, 307359360, 6849722880, 155831195520, 3636061228800, 87265469491200, 2157837063782400, 55024845126451200, 1447576694865100800, 39291367432052736000, 1100158288097476608000, 31767070568814637056000
Offset: 6

Views

Author

N. J. A. Sloane

Keywords

References

  • Louis Comtet, Advanced Combinatorics, Reidel, 1974, p. 156.
  • John Riordan, An Introduction to Combinatorial Analysis, Wiley, 1958, p. 44.
  • 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).

Links

Crossrefs

Column 6 of A008297.
Column m=6 of unsigned triangle A111596.
Cf. A001113, A001620, A091725, A099285.

Programs

  • Magma
    [Factorial(n-6)*Binomial(n,6)*Binomial(n-1,5): n in [6..30]]; // G. C. Greubel, May 10 2021
  • Maple
    A001778 := proc(n)
    n!*binomial(n-1,5)/6! ;
end proc:
seq(A001778(n),n=6..30) ; # R. J. Mathar, Jan 06 2021
  • Mathematica
    With[{c=6!},Table[n!Binomial[n-1,5]/c,{n,6,24}]] (* Harvey P. Dale, May 25 2011 *)
  • Sage
    [binomial(n,6)*factorial(n-1)/factorial(5) for n in range(6, 22)] # Zerinvary Lajos, Jul 07 2009

Formula

E.g.f.: ((x/(1-x))^6)/6!.
If we define f(n,i,x) = Sum_{k=i..n} (Sum_{j=i..k} (binomial(k,j)*Stirling1(n,k) *Stirling2(j,i)*x^(k-j) ) ) then a(n) = (-1)^n*f(n,6,-6), (n>=6). - Milan Janjic, Mar 01 2009
D-finite with recurrence (-n+6)*a(n) +n*(n-1)*a(n-1)=0. - R. J. Mathar, Jan 06 2021
From Amiram Eldar, May 02 2022: (Start)
Sum_{n>=6} 1/a(n) = 570*(gamma - Ei(1)) + 1380*e - 2999, where gamma = A001620, Ei(1) = A091725 and e = A001113.
Sum_{n>=6} (-1)^n/a(n) = 15030*(gamma - Ei(-1)) - 9000/e - 8661, where Ei(-1) = -A099285. (End)

Extensions

More terms from Christian G. Bower, Dec 18 2001
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