A145460 -id:A145460 - cp's OEIS frontend

A029651 Central elements of the (1,2)-Pascal triangle A029635.

1, 3, 9, 30, 105, 378, 1386, 5148, 19305, 72930, 277134, 1058148, 4056234, 15600900, 60174900, 232676280, 901620585, 3500409330, 13612702950, 53017895700, 206769793230, 807386811660, 3156148445580, 12350146091400, 48371405524650

Offset: 0

Mohammad K. Azarian

If Y is a fixed 2-subset of a (2n+1)-set X then a(n) is the number of (n+1)-subsets of X intersecting Y. - Milan Janjic, Oct 28 2007

V. N. Smith and L. Shapiro, Catalan numbers, Pascal's triangle and mutators, Congressus Numerant., 205 (2010), 187-197.

G. C. Greubel, Table of n, a(n) for n = 0..1000
Paul Barry, On a Central Transform of Integer Sequences, arXiv:2004.04577 [math.CO], 2020.
C. Bean, M. Tannock and H. Ulfarsson, Pattern avoiding permutations and independent sets in graphs, arXiv:1512.08155 [math.CO], 2015.
Milan Janjic, Two Enumerative Functions
Mark C. Wilson, Asymptotics for generalized Riordan arrays, International Conference on Analysis of Algorithms DMTCS proc. AD. Vol. 323. 2005. (However, the asymptotics given there on p. 328 for a(n) give different results for me. - _Ralf Stephan_, Dec 28 2013)

Cf. A001700, A143398, A145460.

Essentially a duplicate of A003409.

a := n -> (3/2)*4^n*GAMMA(1/2+n)/(sqrt(Pi)*GAMMA(1+n))-0^n/2;
seq(simplify(a(n)), n=0..24); # Peter Luschny, Dec 16 2015

Join[{1},Table[3*Binomial[2n-1,n],{n,30}]] (* Harvey P. Dale, Aug 11 2015 *)

concat([1], for(n=1, 50, print1(3*binomial(2*n-1,n), ", "))) \\ G. C. Greubel, Jan 23 2017

a(n) = 3 * binomial(2n-1, n) (n>0). - Len Smiley, Nov 03 2001
a(n) = 3*A001700(n-1), (n>=1).
G.f.: (1+xC(x))/(1-2xC(x)), C(x) the g.f. of A000108. - Paul Barry, Dec 17 2004
a(n) = A003409(n), n>0. - R. J. Mathar, Oct 23 2008
a(n) = Sum_{k=0..n} A039599(n,k)*A000034(k). - Philippe Deléham, Oct 29 2008
a(n) = (3/2)*4^n*Gamma(1/2+n)/(sqrt(Pi)*Gamma(1+n))-0^n/2. - Peter Luschny, Dec 16 2015
a(n) ~ (3/2)*4^n*(1-(1/8)/n+(1/128)/n^2+(5/1024)/n^3-(21/32768)/n^4)/sqrt(n*Pi). - Peter Luschny, Dec 16 2015
a(n) = 2^(1-n)*Sum_{k=0..n} binomial(k+n,k)*binomial(2*n-1,n-k), n>0, a(0)=1. - Vladimir Kruchinin, Nov 23 2016
E.g.f.: (3*exp(2*x)*BesselI(0,2*x) - 1)/2. - Ilya Gutkovskiy, Nov 23 2016
a(n) = A143398(2n,n) = A145460(2n,n). - Alois P. Heinz, Sep 09 2018
a(n) = [x^n] C(-x)^(-3*n), where C(x) = (1 - sqrt(1 - 4*x))/(2*x) is the g.f. of the Catalan numbers A000108. - Peter Bala, Oct 16 2024

More terms from David W. Wilson

A143398 Triangle T(n,k) = number of forests of labeled rooted trees of height at most 1, with n labels, where each root contains k labels, n>=0, 0<=k<=n.

Original entry on oeis.org

1, 0, 1, 0, 3, 1, 0, 10, 3, 1, 0, 41, 9, 4, 1, 0, 196, 40, 10, 5, 1, 0, 1057, 210, 30, 15, 6, 1, 0, 6322, 1176, 175, 35, 21, 7, 1, 0, 41393, 7273, 1176, 105, 56, 28, 8, 1, 0, 293608, 49932, 7084, 756, 126, 84, 36, 9, 1, 0, 2237921, 372060, 42120, 6510, 378, 210, 120, 45, 10, 1

Offset: 0

Alois P. Heinz, Aug 12 2008

			T(4,2) = 9: 3->{1,2}<-4, 2->{1,3}<-4, 2->{1,4}<-3, 1->{2,3}<-4, 1->{2,4}<-3, 1->{3,4}<-2, {1,2}{3,4}, {1,3}{2,4}, {1,4}{2,3}.
Triangle begins:
  1;
  0,   1;
  0,   3,  1;
  0,  10,  3,  1;
  0,  41,  9,  4,  1;
  0, 196, 40, 10,  5,  1;
  ...

Alois P. Heinz, Rows n = 0..140, flattened
Index entries for sequences related to rooted trees

Columns k=0-9 give: A000007, A000248, A133189, A145453, A145454, A145455, A145456, A145457, A145458, A145459.
Main diagonal gives A000012.
Row sums give A143406.
T(2n,n) gives A029651.
Cf. A000142, A145460, A351703.

u:= (n, k)-> `if`(k=0, 0, floor(n/k)):
T:= (n, k)-> n! *add(i^(n-k*i)/ ((n-k*i)! *i! *k!^i), i=0..u(n, k)):
seq(seq(T(n, k), k=0..n), n=0..12);

t[n_, n_] = 1; t[, 0] = 0; t[n, k_] := n!*Sum[i^(n-k*i)/((n-k*i)!*i!*k!^i), {i, 0, n/k}]; Table[t[n, k], {n, 0, 12}, {k, 0, n}] // Flatten (* Jean-François Alcover, Dec 19 2013 *)

u(n,k) = if(k==0, 0, n\k);
T(n, k) = n!*sum(j=0, u(n, k), j^(n-k*j)/(k!^j*j!*(n-k*j)!)); \\ Seiichi Manyama, May 13 2022

T(n,k) = n! * Sum_{i=0..u(n,k)} i^(n-k*i)/((n-k*i)!*i!*k!^i) with u(n,k) = 0 if k=0 and u(n,k) = floor(n/k) else.

A274804 The exponential transform of sigma(n).

Original entry on oeis.org

1, 1, 4, 14, 69, 367, 2284, 15430, 115146, 924555, 7991892, 73547322, 718621516, 7410375897, 80405501540, 914492881330, 10873902417225, 134808633318271, 1738734267608613, 23282225008741565, 323082222240744379, 4638440974576329923, 68794595993688306903

Offset: 0

Johannes W. Meijer, Jul 27 2016

nonn

The exponential transform [EXP] transforms an input sequence b(n) into the output sequence a(n). The EXP transform is the inverse of the logarithmic transform [LOG], see the Weisstein link and the Sloane and Plouffe reference. This relation goes by the name of Riddell's formula. For information about the logarithmic transform see A274805. The EXP transform is related to the multinomial transform, see A274760 and the second formula.
The definition of the EXP transform, see the second formula, shows that n >= 1. To preserve the identity LOG[EXP[b(n)]] = b(n) for n >= 0 for a sequence b(n) with offset 0 the shifted sequence b(n-1) with offset 1 has to be used as input for the exponential transform, otherwise information about b(0) will be lost in transformation.
In the a(n) formulas, see the examples, the multinomial coefficients A178867 appear.
We observe that a(0) = 1 and provides no information about any value of b(n), this notwithstanding it is customary to start the a(n) sequence with a(0) = 1.
The Maple programs can be used to generate the exponential transform of a sequence. The first program uses a formula found by Alois P. Heinz, see A007446 and the first formula. The second program uses the definition of the exponential transform, see the Weisstein link and the second formula. The third program uses information about the inverse of the exponential transform, see A274805.
Some EXP transform pairs are, n >= 1: A000435(n) and A065440(n-1); 1/A000027(n) and A177208(n-1)/A177209(n-1); A000670(n) and A075729(n-1); A000670(n-1) and A014304(n-1); A000045(n) and A256180(n-1); A000290(n) and A033462(n-1); A006125(n) and A197505(n-1); A053549(n) and A198046(n-1); A000311(n) and A006351(n); A030019(n) and A134954(n-1); A038048(n) and A053529(n-1); A193356(n) and A003727(n-1).

			Some a(n) formulas, see A178867:
a(0) = 1
a(1) = x(1)
a(2) = x(1)^2 + x(2)
a(3) = x(1)^3 + 3*x(1)*x(2) + x(3)
a(4) = x(1)^4 + 6*x(1)^2*x(2) + 4*x(1)*x(3) + 3*x(2)^2 + x(4)
a(5) = x(1)^5 + 10*x(1)^3*x(2) + 10*x(1)^2*x(3) + 15*x(1)*x(2)^2 + 5*x(1)*x(4) + 10*x(2)*x(3) + x(5)

Frank Harary and Edgar M. Palmer, Graphical Enumeration, 1973.
Robert James Riddell, Contributions to the theory of condensation, Dissertation, University of Michigan, Ann Arbor, 1951.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, 1995, pp. 18-23.

Alois P. Heinz, Table of n, a(n) for n = 0..531
M. Bernstein and N. J. A. Sloane, Some Canonical Sequences of Integers, Linear Algebra and its Applications, Vol. 226-228 (1995), pp. 57-72. Erratum 320 (2000), 210. [Link to arXiv version]
M. Bernstein and N. J. A. Sloane, Some canonical sequences of integers, Linear Alg. Applications, 226-228 (1995), 57-72; erratum 320 (2000), 210. [Link to Lin. Alg. Applic. version together with omitted figures]
N. J. A. Sloane, Transforms.
Eric W. Weisstein MathWorld, Exponential Transform.

Cf. A274805, A274760, A178867, A000203, A007446.
Cf. A177208, A177209, A006351, A197505, A144180, A256180, A033462, A198046, A134954, A145460, A188489, A005432, A029725, A124213, A002801.
Cf. A036040, another version.

nmax:=21: with(numtheory): b := proc(n): sigma(n) end: a:= proc(n) option remember; if n=0 then 1 else add(binomial(n-1, j-1) * b(j) *a(n-j), j=1..n) fi: end: seq(a(n), n=0..nmax); # End first EXP program.
nmax:= 21: with(numtheory): b := proc(n): sigma(n) end: t1 := exp(add(b(n)*x^n/n!, n=1..nmax+1)): t2 := series(t1, x, nmax+1): a := proc(n): n!*coeff(t2, x, n) end: seq(a(n), n=0..nmax); # End second EXP program.
nmax:=21: with(numtheory): b := proc(n): sigma(n) end: f := series(log(1+add(q(n)*x^n/n!, n=1..nmax+1)), x, nmax+1): d := proc(n): n!*coeff(f, x, n) end: a(0):=1: q(0):=1: a(1):=b(1): q(1):=b(1): for n from 2 to nmax+1 do q(n) := solve(d(n)-b(n), q(n)): a(n):=q(n): od: seq(a(n), n=0..nmax); # End third EXP program.

a[0] = 1; a[n_] := a[n] = Sum[Binomial[n-1, j-1]*DivisorSigma[1, j]*a[n-j], {j, 1, n}]; Table[a[n], {n, 0, 30}] (* Jean-François Alcover, Feb 22 2017 *)
nmax = 20; CoefficientList[Series[Exp[Sum[DivisorSigma[1, k]*x^k/k!, {k, 1, nmax}]], {x, 0, nmax}], x] * Range[0, nmax]! (* Vaclav Kotesovec, Jun 08 2021 *)

a(n) = Sum_{j=1..n} (binomial(n-1,j-1) * b(j) * a(n-j)), n >= 1 and a(0) = 1, with b(n) = A000203(n) = sigma(n).

E.g.f.: exp(Sum_{n >= 1} b(n)*x^n/n!) with b(n) = sigma(n) = A000203(n).

A279636 Square array A(n,k), n>=0, k>=0, read by antidiagonals, where column k is the exponential transform of the k-th powers.

Original entry on oeis.org

1, 1, 1, 1, 1, 2, 1, 1, 3, 5, 1, 1, 5, 10, 15, 1, 1, 9, 22, 41, 52, 1, 1, 17, 52, 125, 196, 203, 1, 1, 33, 130, 413, 836, 1057, 877, 1, 1, 65, 340, 1445, 3916, 6277, 6322, 4140, 1, 1, 129, 922, 5261, 19676, 41077, 52396, 41393, 21147, 1, 1, 257, 2572, 19685, 104116, 288517, 481384, 479593, 293608, 115975

Offset: 0

Alois P. Heinz, Dec 16 2016

			Square array A(n,k) begins:
:   1,    1,    1,     1,      1,       1,        1, ...
:   1,    1,    1,     1,      1,       1,        1, ...
:   2,    3,    5,     9,     17,      33,       65, ...
:   5,   10,   22,    52,    130,     340,      922, ...
:  15,   41,  125,   413,   1445,    5261,    19685, ...
:  52,  196,  836,  3916,  19676,  104116,   572036, ...
: 203, 1057, 6277, 41077, 288517, 2133397, 16379797, ...

Alois P. Heinz, Antidiagonals n = 0..140, flattened
Wikipedia, Kronecker delta

Columns k=0-10 give: A000110, A000248, A033462, A279358, A279637, A279638, A279639, A279640, A279641, A279642, A279643.
Rows n=0+1,2 give: A000012, A000051.
Main diagonal gives A279644.
Cf. A145460.

egf:= k-> exp(exp(x)*add(Stirling2(k, j)*x^j, j=0..k)-`if`(k=0, 1, 0)):
A:= (n, k)-> n!*coeff(series(egf(k), x, n+1), x, n):
seq(seq(A(n, d-n), n=0..d), d=0..12);
# second Maple program:
A:= proc(n, k) option remember; `if`(n=0, 1,
      add(binomial(n-1, j-1)*j^k*A(n-j, k), j=1..n))
    end:
seq(seq(A(n, d-n), n=0..d), d=0..12);

A[n_, k_] := A[n, k] = If[n==0, 1, Sum[Binomial[n-1, j-1]*j^k*A[n-j, k], {j, 1, n}]]; Table[A[n, d-n], {d, 0, 12}, {n, 0, d}] // Flatten (* Jean-François Alcover, Feb 19 2017, translated from Maple *)

E.g.f. of column k: exp(exp(x)*(Sum_{j=0..k} Stirling2(n,j)*x^j) - delta_{0,k}).

A145453 Exponential transform of binomial(n,3) = A000292(n-2).

Original entry on oeis.org

1, 0, 0, 1, 4, 10, 30, 175, 1176, 7084, 42120, 286605, 2270180, 19213766, 166326524, 1497096055, 14374680880, 147259920760, 1582837679056, 17659771122969, 204674606377140, 2473357218561250, 31148510170120420, 407154732691440811, 5504706823227724904

Offset: 0

Alois P. Heinz, Oct 10 2008

nonn

a(n) is the number of ways of placing n labeled balls into indistinguishable boxes, where in each filled box 3 balls are seen at the top.

a(n) is also the number of forests of labeled rooted trees of height at most 1, with n labels, where each root contains 3 labels.

Seiichi Manyama, Table of n, a(n) for n = 0..530 (terms 0..200 from Alois P. Heinz)
N. J. A. Sloane, Transforms
Index entries for sequences related to rooted trees

3rd column of A145460, A143398.

Cf. A292889, A292910.

a:= proc(n) option remember; `if`(n=0, 1,
      add(binomial(n-1, j-1) *binomial(j,3) *a(n-j), j=1..n))
    end:
seq(a(n), n=0..30);

Table[Sum[BellY[n, k, Binomial[Range[n], 3]], {k, 0, n}], {n, 0, 25}] (* Vladimir Reshetnikov, Nov 09 2016 *)

E.g.f.: exp(exp(x)*x^3/3!).

A292948 Square array A(n,k), n>=0, k>=0, read by antidiagonals, where A(0,k) = 1 and A(n,k) = (-1)^(k+1) * Sum_{i=0..n-1} (-1)^i * binomial(n-1,i) * binomial(i+1,k) * A(n-1-i,k) for n > 0.

Original entry on oeis.org

1, 1, -1, 1, 1, 2, 1, 0, -1, -5, 1, 0, 1, -2, 15, 1, 0, 0, -3, 9, -52, 1, 0, 0, 1, 9, -4, 203, 1, 0, 0, 0, -4, -40, -95, -877, 1, 0, 0, 0, 1, 10, 210, 414, 4140, 1, 0, 0, 0, 0, -5, -10, -1176, 49, -21147, 1, 0, 0, 0, 0, 1, 15, -105, 7273, -10088, 115975, 1, 0, 0

Offset: 0

Seiichi Manyama, Sep 27 2017

			Square array begins:
    1,  1,  1,  1, 1, ...
   -1,  1,  0,  0, 0, ...
    2, -1,  1,  0, 0, ...
   -5, -2, -3,  1, 0, ...
   15,  9,  9, -4, 1, ...

Seiichi Manyama, Antidiagonals n = 0..139, flattened

Columns k=0-5 give: A292935, A003725, A292909, A292910, A292912, A292950.
Rows n=0 gives A000012.
Main diagonal gives A000012.
Cf. A145460.

def ncr(n, r)
  return 1 if r == 0
  (n - r + 1..n).inject(:*) / (1..r).inject(:*)
end
def A(k, n)
  ary = [1]
  (1..n).each{|i| ary << (-1) ** (k % 2 + 1) * (0..i - 1).inject(0){|s, j| s + (-1) ** (j % 2) * ncr(i - 1, j) * ncr(j + 1, k) * ary[i - 1 - j]}}
  ary
end
def A292948(n)
  a = []
  (0..n).each{|i| a << A(i, n - i)}
  ary = []
  (0..n).each{|i|
    (0..i).each{|j|
      ary << a[i - j][j]
    }
  }
  ary
end
p A292948(20)

A133189 Number of simple directed graphs on n labeled nodes consisting only of some cycle graphs C_2 and nodes not part of a cycle having directed edges to both nodes in exactly one cycle.

Original entry on oeis.org

1, 0, 1, 3, 9, 40, 210, 1176, 7273, 49932, 372060, 2971540, 25359411, 230364498, 2215550428, 22460391240, 239236043985, 2669869110856, 31134833803728, 378485082644400, 4786085290280275, 62838103267148790, 855122923978737876, 12042364529117844328

Offset: 0

Alois P. Heinz, Dec 17 2007

nonn

			a(3) = 3, because there are 3 graphs of the given kind for 3 labeled nodes: 3->1<->2<-3,  2->1<->3<-2,  1->2<->3<-1.

A. P. Heinz (1990). Analyse der Grenzen und Möglichkeiten schneller Tableauoptimierung. PhD Thesis, Albert-Ludwigs-Universität Freiburg, Freiburg i. Br., Germany.

Alois P. Heinz, Table of n, a(n) for n = 0..530
Eric Weisstein's World of Mathematics, Directed Graph
Eric Weisstein's World of Mathematics, Cycle Graph

Cf. A006882, A007318, A135458, A135429.

2nd column of A145460, A143398.

a:= proc(n) option remember; add(binomial(n, k+k)*
      doublefactorial(k+k-1) *k^(n-k-k), k=0..floor(n/2))
    end:
seq(a(n), n=0..30);
# second Maple program:
a:= proc(n) option remember; `if`(n=0, 1, add(
      binomial(n-1, j-1) *binomial(j, 2) *a(n-j), j=1..n))
    end:
seq(a(n), n=0..30);  # Alois P. Heinz, Mar 16 2015

nn=20;Range[0,nn]!CoefficientList[Series[Exp[Exp[x]x^2/2],{x,0,nn}],x]  (* Geoffrey Critzer, Nov 23 2012 *)
Table[Sum[BellY[n, k, Binomial[Range[n], 2]], {k, 0, n}], {n, 0, 25}] (* Vladimir Reshetnikov, Nov 09 2016 *)

a(n) = Sum_{k=0..floor(n/2)} binomial(n,2*k) * A006882(2*k-1) * k^(n-2*k).

E.g.f.: exp(exp(x)*x^2/2). - Geoffrey Critzer, Nov 23 2012

A145454 Exponential transform of binomial(n,4) = A000332.

Original entry on oeis.org

1, 0, 0, 0, 1, 5, 15, 35, 105, 756, 6510, 46530, 283470, 1667380, 11457446, 99776040, 969295145, 9298091180, 86154691680, 804769174536, 8052676029420, 88489327173660, 1038440150703340, 12501684521410700, 151866259113256611

Offset: 0

Alois P. Heinz, Oct 10 2008

nonn

a(n) is the number of ways of placing n labeled balls into indistinguishable boxes, where in each filled box 4 balls are seen at the top.

a(n) is also the number of forests of labeled rooted trees of height at most 1, with n labels, where each root contains 4 labels.

Alois P. Heinz, Table of n, a(n) for n = 0..500
N. J. A. Sloane, Transforms
Index entries for sequences related to rooted trees

4th column of A145460, A143398.

a:= proc(n) option remember; `if`(n=0, 1, add(
      binomial(n-1, j-1) *binomial(j,4) *a(n-j), j=1..n))
    end:
seq(a(n), n=0..30);

Table[Sum[BellY[n, k, Binomial[Range[n], 4]], {k, 0, n}], {n, 0, 25}] (* Vladimir Reshetnikov, Nov 09 2016 *)

E.g.f.: exp(exp(x)*x^4/4!).

A293015 Square array A(n,k), n>=0, k>=0, read by antidiagonals, where A(0,k) = 1 and A(n,k) = - Sum_{i=0..n-1} binomial(n-1,i) * binomial(i+1,k) * A(n-1-i,k) for n > 0.

Original entry on oeis.org

1, 1, -1, 1, -1, 0, 1, 0, -1, 1, 1, 0, -1, 2, 1, 1, 0, 0, -3, 9, -2, 1, 0, 0, -1, -3, 4, -9, 1, 0, 0, 0, -4, 20, -95, -9, 1, 0, 0, 0, -1, -10, 150, -414, 50, 1, 0, 0, 0, 0, -5, -10, 504, 49, 267, 1, 0, 0, 0, 0, -1, -15, 105, -343, 10088, 413, 1, 0, 0, 0, 0, 0, -6

Offset: 0

Seiichi Manyama, Sep 28 2017

			Square array begins:
    1,  1,  1,  1,  1, ...
   -1, -1,  0,  0,  0, ...
    0, -1, -1,  0,  0, ...
    1,  2, -3, -1,  0, ...
    1,  9, -3, -4, -1, ...

Seiichi Manyama, Antidiagonals n = 0..139, flattened

Columns k=0-4 give: A000587, A292952, A292953, A292954, A292955.
Rows n=0 gives A000012.
Cf. A145460.

A145455 Exponential transform of C(n,5) = A000389.

Original entry on oeis.org

1, 0, 0, 0, 0, 1, 6, 21, 56, 126, 378, 3234, 34056, 289575, 2020018, 12237225, 70634928, 468041756, 4190274648, 45557515620, 499503011496, 5121432107757, 49183015347774, 462331794763069, 4579813478536296, 50959878972009546

Offset: 0

Alois P. Heinz, Oct 10 2008

nonn

a(n) is the number of ways of placing n labeled balls into indistinguishable boxes, where in each filled box 5 balls are seen at the top.

a(n) is also the number of forests of labeled rooted trees of height at most 1, with n labels, where each root contains 5 labels.

Alois P. Heinz, Table of n, a(n) for n = 0..500
N. J. A. Sloane, Transforms
Index entries for sequences related to rooted trees

5th column of A145460, A143398.

a:= proc(n) option remember; `if`(n=0, 1,
      add(binomial(n-1, j-1) *binomial(j,5) *a(n-j), j=1..n))
    end:
seq(a(n), n=0..30);

With[{nmax = 50}, CoefficientList[Series[Exp[Exp[x]*x^5/5!], {x, 0, nmax}], x]] (* G. C. Greubel, Nov 21 2017 *)

x='x+O('x^30); Vec(serlaplace(exp(exp(x)*x^5/5!))) \\ G. C. Greubel, Nov 21 2017

E.g.f.: exp(exp(x)*x^5/5!).

cp's OEIS Frontend

A029651 Central elements of the (1,2)-Pascal triangle A029635.

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A143398 Triangle T(n,k) = number of forests of labeled rooted trees of height at most 1, with n labels, where each root contains k labels, n>=0, 0<=k<=n.

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Maple

Mathematica

PARI

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A274804 The exponential transform of sigma(n).

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Maple

Mathematica

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A279636 Square array A(n,k), n>=0, k>=0, read by antidiagonals, where column k is the exponential transform of the k-th powers.

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Maple

Mathematica

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A145453 Exponential transform of binomial(n,3) = A000292(n-2).

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Maple

Mathematica

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A292948 Square array A(n,k), n>=0, k>=0, read by antidiagonals, where A(0,k) = 1 and A(n,k) = (-1)^(k+1) * Sum_{i=0..n-1} (-1)^i * binomial(n-1,i) * binomial(i+1,k) * A(n-1-i,k) for n > 0.

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Ruby

A133189 Number of simple directed graphs on n labeled nodes consisting only of some cycle graphs C_2 and nodes not part of a cycle having directed edges to both nodes in exactly one cycle.

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