A073146 -id:A073146 - cp's OEIS frontend

A098696 Main diagonal of array in A073146.

1, 4, 104, 6544, 765344, 143639104, 39511297664, 14979797833984, 7487442077817344, 4770988535512474624, 3774873839360539879424, 3630982576832133263233024, 4172729918808369709126098944

Offset: 0

Ralf Stephan, Sep 23 2004

nonn

Diagonal of Euler-Seidel matrix with start sequence A000670.

Cf. A000670, A073146.

Table[Sum[Binomial[n, k] * Sum[StirlingS2[2*n-k, j]*j!, {j, 0, 2*n-k}],{k,0,n}],{n,0,20}] (* Vaclav Kotesovec, May 03 2015 *)

a(n) = Sum_{k=0..n} binomial(n, k) * A000670(2*n-k).
a(n) ~ (2*n)! / (sqrt(2) * (log(2))^(2*n+1)). - Vaclav Kotesovec, May 03 2015
a(n) = Sum_{k>=0} (k*(k + 1))^n/2^(k+1). - Ilya Gutkovskiy, Jun 29 2019

Offset corrected by Vaclav Kotesovec, May 03 2015

A162509 Row sums of the absolute values of a triangular array related to the Bernoulli numbers.

Original entry on oeis.org

1, 1, 4, 20, 124, 932, 8284, 85220, 997084, 13082852, 190320604, 3040770020, 52937870044, 997533561572, 20228969244124, 439283696014820, 10170742982007004, 250110224694309092, 6510327792455418844, 178832105312143131620, 5169772417850111583964

Offset: 0

Peter Luschny, Jul 05 2009

nonn

Let T(n,k) = sum_{v=0..k} (-1)^v*v*binomial(k,v)*(v+1)^(n-1) for n >= 1, k >= 1 and additionally T(0,0) = 1. Then a(n) = sum_{k=0..n} abs(T(n,k)).
a(n) = A073146(n,n-1) for n >= 1.
a(n) appears to be the total number of subsets over all chains of the poset on the powerset of {1,2,...,n-1} ordered by set inclusion.  That is,  a(n) = Sum_{k=0..n} A038719(n,k)*(k+1).  For example a(2)=4 because there are three chains: {}; {1}; {},{1}; and there are 4 total subsets. - Geoffrey Critzer, Nov 28 2014

Bruno Berselli, Table of n, a(n) for n = 0..100

Cf. A162508, A073146.

A162508 := proc(n,k) local v; if n=0 and k=0 then 1 else
add((-1)^v*v*binomial(k,v)*(v+1)^(n-1),v=0..k) fi end:
a := proc(n) local k; add(abs(A162508(n,k)),k=0..n) end:

t[0, 0] = 1; t[n_, k_] := Sum[(-1)^v*v*Binomial[k, v]*(v+1)^(n-1), {v, 0, k}]; a[n_] := Sum[Abs[t[n, k]], {k, 0, n}]; Table[a[n], {n, 0, 16}] (* Jean-François Alcover, Jun 28 2013 *)

def A162509(n):
    return add(abs(A162508(n, k)) for k in (0..n))
[A162509(n) for n in (0..20)] # Peter Luschny, Jul 21 2014

a(n+1)=Sum_{k, 0<=k<=n} A199400(n,k) = Sum_{k, 0<=k<=n} A199335(n,k)*2^k. - Philippe Deléham, Nov 06 2011
G.f.: 1+x/(1-4x/(1-x/(1-6x/(1-2x/(1-8x/(1-3x/(1-10x/(1-4x/1-....)))))))) (continued fraction). - Philippe Deléham, Nov 22 2011
G.f.: 1 + x/Q(0), where Q(k) = 1 - x*(3*k+4) - 2*x^2*(k+1)*(k+2)/Q(k+1); (continued fraction). - Sergei N. Gladkovskii, Oct 03 2013
a(n + 1) = sum {k >= 0} (k*(k + 1)^n)/2^(k + 1) for n >= 0. Comparison with the formula A000670(n) = sum {k >= 0} (k^n)/2^(k + 1) yields a(n + 1) = sum {k = 0..n} binomial(n,k)*A000670(k + 1). - Peter Bala, Jul 21 2014
a(n) ~ n! / log(2)^(n+1). - Vaclav Kotesovec, Apr 17 2018

A122101 T(n,k) = Sum_{i=0..k} (-1)^(k-i)binomial(k,i)A000670(n-k+i).

Original entry on oeis.org

1, 1, 0, 3, 2, 2, 13, 10, 8, 6, 75, 62, 52, 44, 38, 541, 466, 404, 352, 308, 270, 4683, 4142, 3676, 3272, 2920, 2612, 2342, 47293, 42610, 38468, 34792, 31520, 28600, 25988, 23646, 545835, 498542, 455932, 417464, 382672, 351152, 322552, 296564, 272918

Offset: 0

Vladeta Jovovic, Oct 18 2006

			Triangle begins as:
     1;
     1,    0;
     3,    2,    2;
    13,   10,    8,    6;
    75,   62,   52,   44,   38;
   541,  466,  404,  352,  308,  270;
  4683, 4142, 3676, 3272, 2920, 2612, 2342;
  ...

G. C. Greubel, Rows n = 0..100 of triangle, flattened

Columns k=0-1 give: A000670, A232472.
Row sums give A089677(n+1).
Main diagonal gives A052841.
T(2n,n) gives A340837.
Cf. A005649, A069321, A073146.

A000670:= function(n)
     return Sum([0..n], i-> Factorial(i)*Stirling2(n,i) ); end;
T:= function(n,k)
    return Sum([0..k], j-> (-1)^(k-j)*Binomial(k, j)*A000670(n-k+j) ); end;
Flat(List([0..10], n-> List([0..n], k-> T(n,k) ))); # G. C. Greubel, Oct 02 2019

A000670:= func< n | &+[Factorial(k)*StirlingSecond(n,k): k in [0..n]] >;
[(&+[(-1)^(k-j)*Binomial(k,j)*A000670(n-k+j): j in [0..k]]): k in [0..n], n in [0..10]]; // G. C. Greubel, Oct 02 2019

T:= (n, k)-> k!*(n-k)!*coeff(series(coeff(series(exp(-y)/
        (2-exp(x+y)), y, k+1), y, k), x, n-k+1), x, n-k):
seq(seq(T(n, k), k=0..n), n=0..12);  # Alois P. Heinz, Oct 02 2019
# second Maple program:
b:= proc(n) option remember; `if`(n<2, 1,
      add(b(n-j)*binomial(n, j), j=1..n))
    end:
T:= (n, k)-> add(binomial(k, j)*(-1)^j*b(n-j), j=0..k):
seq(seq(T(n, k), k=0..n), n=0..12);  # Alois P. Heinz, Oct 02 2019

A000670[n_]:= If[n==0,1,Sum[k! StirlingS2[n, k], {k, n}]]; T[n_, k_]:= Sum[(-1)^(k-j)*Binomial[k, j]*A000670[n-k+j], {j,0,k}]; Table[T[n, k], {n,0,10}, {k,0,n}]//Flatten (* G. C. Greubel, Oct 02 2019 *)

A000670(n) = sum(k=0,n, k!*stirling(n,k,2));
T(n,k) = sum(j=0,k, (-1)^(k-j)*binomial(k, j)*A000670(n-k+j));
for(n=0,10, for(k=0,n, print1(T(n,k), ", "))) \\ G. C. Greubel, Oct 02 2019

def A000670(n): return sum(factorial(k)*stirling_number2(n,k) for k in (0..n))
def T(n,k): return sum((-1)^(k-j)*binomial(k, j)*A000670(n-k+j) for j in (0..k))
[[T(n,k) for k in (0..n)] for n in (0..10)]

Doubly-exponential generating function: Sum_{n, k} a(n-k,k) x^n/n! y^k/k! = exp(-y)/(2-exp(x+y)).

A162508 A binomial sum of powers related to the Bernoulli numbers, triangular array, read by rows.

Original entry on oeis.org

-1, -2, 2, -4, 10, -6, -8, 38, -54, 24, -16, 130, -330, 336, -120, -32, 422, -1710, 3000, -2400, 720, -64, 1330, -8106, 21840, -29400, 19440, -5040, -128, 4118, -36414, 141624, -285600, 312480, -176400, 40320

Offset: 1

Peter Luschny, Jul 05 2009

T(n,k) = sum_{v=0..k} (-1)^v*v*binomial(k,v)*(v+1)^(n-1)
for n >= 1, k >= 1; by convention T(0,0) = 1.
Gives a representation of the Bernoulli numbers B_{n} = B_{n}(1) (with B_1 = 1/2)
B_{n} = sum_{j=0..n} sum_{k=0..j} T(j,k)/(k+1)
T(n,1) = -2^(n-1) (n>=1)
T(n,n) = (-1)^n*n! (n>=1)
sum_{k=0..n} T(n,k) = -A000007(n-1) = -1,0,0,0,0,... (n>=1)
sum_{k=0..n} abs(T(n,k)) = A162509(n) = A073146(n,n-1) (n>=1)
sum_{k=0..n} T(n,k)/(k+1) = Bernoulli(n,1)-Bernoulli(n-1,1) (n>=1)
numer(sum(T(n,k)/(k+1),k=0..n)) = A051716(n) (n>=0)
denom(sum(T(n,k)/(k+1),k=0..n)) = A051717(n) (n>=0)
Contribution from Peter Luschny, Jul 08 2009: (Start)
More generally, define the polynomials (assume p[0,0](x)=1 and 0^0=1)
p[n,k](x) = sum_{v=0..k} (-1)^v*v*binomial(k,v)*(v+1+x)^(n-1)
[1], [0, -1], [0, -2-x, 2], [0, -4-4x-x^2, 10+4x, -6], ...
then T(n,k)=p[n,k](0) and (-1)^k*k!*Stirling2(n,k)=p[n,k](-1) (cf. A019538).
Assume now k >= 1 and read by rows. Then
p[n,k](1) = -1,-3,2,-9,14,-6,-27,74,-72,24,-81,350,-582,432,-120,...
(-1)^n*(-2)^(n-k)*p[n,k](-1/2))=1,3,2,9,16,6,27,98,90,24,81,544,924,576,120,..
(-1)^n*(-2)^(n-k)*p[n,k](-3/2))=1,1,2,1,8,6,1,26,54,24,1,80,348,384,120,... (End)
Variant of A199400.

			For n >= 1, k >= 1:
..................... -1
................... -2, 2
................. -4, 10, -6
.............. -8, 38, -54, 24
......... -16, 130, -330, 336, -120
..... -32, 422, -1710, 3000, -2400, 720
-64, 1330, -8106, 21840, -29400, 19440, -5040

Vincenzo Librandi, Rows n = 1..50, flattened

Cf. A162509, A073146, A051716, A051717, A199400. A019538, A028246, A038719.

T := proc(n,k) local v; if n=0 and k=0 then 1 else
add((-1)^v*v*binomial(k,v)*(v+1)^(n-1),v=0..k) fi end:
# Peter Bala's e.g.f. assuming offset 0:
egf := (x, z) -> -((1-x)/exp(z) + x)^(-2):
ser := series(egf(x, z), z, 10): coz := n -> n!*coeff(ser, z, n):
row := n -> seq(coeff(coz(n), x, k), k = 0..n):
seq(print(row(n)), n = 0..9); # Peter Luschny, Jan 28 2021

t[n_, k_] := Sum[(-1)^v*v*Binomial[k, v]*(v + 1)^(n - 1), {v, 0, k}]; Table[t[n, k], {n, 1, 8}, {k, 1, n}] // Flatten (* Jean-François Alcover, Jul 26 2013, after Maple *)

def A162508(n, k):
    if n==0 and k==0: return 1
    return add((-1)^v*v*binomial(k, v)*(v+1)^(n-1) for v in (0..k))
for n in (1..8): [A162508(n, k) for k in (1..n)] # Peter Luschny, Jul 21 2014

From Peter Bala, Jul 21 2014: (Start)
T(n,k) = (-1)^k*k!*( Stirling2(n+1,k+1) - Stirling2(n,k+1) ), 1 <= k <= n.
T(n,k) = (-1)^k*(k + 1)*A038719(n+1,k+1).
E.g.f.: - B(-x,z)^2, where B(x,z) = 1/((1 + x)*exp(-z) - x) = 1 + (1 + x)*z + (1 + 3*x + 2*x^2)*z^2/2! + ... is an e.g.f. for A028246 (with an offset of 0).
Recurrence: T(n,k) = (k + 1)*T(n-1,k) - k*T(n-1,k-1).
The unsigned version of the triangle equals the matrix product A007318*A019538.
Assuming this triangle is a signed version of A199400 then the n-th row polynomial R(n,x) = 1/(1 - x)*( sum {k = 1..inf} k*(k + 1)^(n-1)*(x/(x - 1))^k ), valid for x in the open interval (-inf, 1/2). (End)

More terms from Philippe Deléham, Nov 05 2011

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A098696 Main diagonal of array in A073146.

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A162509 Row sums of the absolute values of a triangular array related to the Bernoulli numbers.

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A122101 T(n,k) = Sum_{i=0..k} (-1)^(k-i)*binomial(k,i)*A000670(n-k+i).

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A162508 A binomial sum of powers related to the Bernoulli numbers, triangular array, read by rows.

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A122101 T(n,k) = Sum_{i=0..k} (-1)^(k-i)binomial(k,i)A000670(n-k+i).