A195205 -id:A195205 - cp's OEIS frontend

A079641 Matrix product of Stirling2-triangle A008277(n,k) and unsigned Stirling1-triangle |A008275(n,k)|.

1, 2, 1, 6, 6, 1, 26, 36, 12, 1, 150, 250, 120, 20, 1, 1082, 2040, 1230, 300, 30, 1, 9366, 19334, 13650, 4270, 630, 42, 1, 94586, 209580, 166376, 62160, 11900, 1176, 56, 1, 1091670, 2562354, 2229444, 952728, 220500, 28476, 2016, 72, 1, 14174522

Offset: 1

Vladeta Jovovic, Jan 30 2003

Triangle T(n,k), 1<=k<=n, read by rows, given by (0, 2, 1, 4, 2, 6, 3, 8, 4, 10, 5, ...) DELTA (1, 0, 1, 0, 1, 0, 1, 0, 1, 0, ...) where DELTA is the operator defined in A084938. - Philippe Deléham, Dec 22 2011
Subtriangle of triangle in A129062. - Philippe Deléham, Feb 17 2013
Also the Bell transform of A000629. For the definition of the Bell transform see A264428. - Peter Luschny, Jan 26 2016

			Triangle begins:
  1;
  2,1;
  6,6,1;
  26,36,12,1;
  150,250,120,20,1;
  1082,2040,1230,300,30,1;
  ...
Triangle (0,2,1,4,2,6,3,8,4,...) DELTA (1,0,1,0,1,0,1,0,1,...) begins:
  1
  0, 1
  0, 2, 1
  0, 6, 6, 1
  0, 26, 36, 12, 1
  0, 150, 250, 120, 20, 1
  0, 1082, 2040, 1230, 300, 30, 1. - _Philippe Deléham_, Dec 22 2011

Nick Early, Canonical Bases for Permutohedral Plates, arXiv:1712.08520 [math.CO], 2017.
Nick Early, Honeycomb tessellations and canonical bases for permutohedral blades, arXiv:1810.03246 [math.CO], 2018.
D. E. Knuth, Convolution polynomials, arXiv:math/9207221 [math.CA], 1992; The Mathematica J., 2 (1992), 67-78.

Cf. A000670 (row sums), A000629 (first column), A195204, A195205. A209849, A129062

# The function BellMatrix is defined in A264428.
# Adds (1, 0, 0, 0, ..) as column 0.
BellMatrix(n -> add((-1)^(n-k)*2^k*k!*combinat:-stirling2(n, k), k=0..n), 9); # Peter Luschny, Jan 26 2016

rows = 10;
t = Table[Sum[(-1)^(n-k)*2^k*k!*StirlingS2[n, k], {k,0,n}], {n, 0, rows}];
T[n_, k_] := BellY[n, k, t];
Table[T[n, k], {n, 1, rows}, {k, 1, n}] // Flatten (* Jean-François Alcover, Jun 22 2018 *)

T(n, k) = Sum_{i=k..n} A008277(n, i) * |A008275(i, k)|.
E.g.f.: (2-exp(x))^(-y). - Vladeta Jovovic, Nov 22 2003
From Peter Bala, Sep 12 2011: (Start)
The row generating polynomials R(n,x) begin R(1,x) = x, R(2,x) = 2*x + x^2, R(3,x) = 6*x + 6*x^2 + x^3 and satisfy the recurrence R(n+1,x) = x*(2*R(n,x+1) - R(n,x)). They form a sequence of binomial type polynomials. In particular, denoting R(n,x) by x^[n] to emphasize the analogies with the monomial polynomials x^n, we have the binomial expansion (x + y)^[n] = Sum_{k = 0..n} binomial(n,k)*x^[n-k]*y^[k].
There is a Dobinski-type formula: exp(-x)*Sum_{k >= 0} (-k)^[n] * x^k/k! = Bell(n,-x). The alternating n-th row entries (-1)^k * T(n,k) are the connection coefficients expressing the polynomial Bell(n,-x) as a linear combination of Bell(k,x), 1 <= k <= n. For example, the list of coefficients of R(4,x) is [26, 36, 12, 1] and we have Bell(4,-x) = -26*Bell(1,x) + 36*Bell(2,x) - 12*Bell(3,x) + Bell(4,x).
The row polynomials also satisfy an analog of the Bernoulli's summation formula for powers of integers, namely, Sum_{k = 1..n} k^[p] = 1/(p+1) * Sum_{k = 0..p} binomial(p+1,k) * B_k * n^[p+1-k], where B_k denotes the Bernoulli numbers. Compare with A195204 and A195205. (End)
Let D be the forward difference operator D(f(x)) = f(x+1) - f(x). Then the n-th row polynomial R(n,x) = 1/f(x) * (x*D)^n(f(x)) with f(x) = 2^x. Cf. A209849. Also cf. A008277, where the row polynomials are given by 1/f(x) * (x*d/dx)^n(f(x)), where now f(x) = exp(x). - Peter Bala, Mar 16 2012
Conjecture: o.g.f. as a continued fraction of Stieltjes type: 1/(1 - x*z/(1 - 2*z/(1 - (x + 1)*z/(1 - 4*z/(1 - (x + 2)*z/(1 - 6*z/(1 - (x + 3)*z/(1 - 8*z/(1 - ... ))))))))) =  1 + x*z + (2*x + x^2)*z^2 + (6*x + 6*x^2 + x^3)*z^3 + .... - Peter Bala, Dec 12 2024

A179929 a(n) = 2^n*A(n, -1/2), A(n, x) the Eulerian polynomials.

Original entry on oeis.org

1, 2, 2, -6, -30, 42, 882, 954, -39870, -203958, 2300562, 29677914, -120958110, -4657703958, -7059175758, 807984602874, 6667870853250, -145556787011958, -2827006784652078, 21703953751815834, 1108558810703202210

Offset: 0

Peter Luschny, Aug 03 2010

sign

Stanislav Sykora, Table of n, a(n) for n = 0..199
OEIS Wiki, Eulerian polynomials

Cf. A000629 = 2^n*A(n, 1/2).

Cf. A212846, A123125, A173018, A195205.

a[n_] := Sum[3^(n-k) (-1)^k k! StirlingS2[n+1, k+1], {k, 0, n}];
Table[a[n], {n, 0, 20}] (* Jean-François Alcover, Jun 17 2019 *)

A179929(n) = {my(s,k,term);
  term = 3^n;s = term*stirling(n+1,1,2);
  for (k=1,n,term *= -k/3;s += term*stirling(n+1,k+1,2););
return(s);} \\ Stanislav Sykora, May 15 2014

From Philippe Deléham, Dec 22 2011: (Start)
a(n) = Sum_{k, 0<=k<=n} A123125(n,k)*(-1)^(n-k)*2^k
a(n) = Sum_{k, 0<=k<=n} A173018(n,k)*2^(n-k)*(-1)^k. (End)
From Peter Bala, Mar 12 2013: (Start)
E.g.f.: 3/(1 + 2exp(-3x)) = 1 + 2x + 2x^2/2! - 6x^3/3! - 30x^4/4! + ....
Recurrence equation: a(n+1) = 3a(n) - Sum_{k=0..n} binomial(n,k) a(k)a(n-k).
(-1)^n*a(n) are the coefficients of a delta operator associated with a sequence of polynomials of binomial type - see A195205. (End)
a(n) ~ n! * 2*3^(n+1)/(Pi^2+(log(2))^2)^(n/2+1) * (Pi*sin(n*arctan(Pi/log(2))) - log(2)*cos(n*arctan(Pi/log(2)))). - Vaclav Kotesovec, Oct 09 2013
From Stanislav Sykora, May 15 2014: (Start)
a(n) = -2*A212846(n) for n > 0.
a(n) = -3^(n+1)*Li(-n, -1/2), with Li(-n, x) = Sum_{k>=0} ((k^n)*(x^k)) the polylogarithm.
a(n) = Sum_{k=0..n} 3^(n-k)*(-1)^k*k!*S(n+1, k+1), S(m, l) the Stirling number of second kind. (End)

A195204 Triangle of coefficients of a sequence of binomial type polynomials.

Original entry on oeis.org

2, 2, 4, 6, 12, 8, 26, 60, 48, 16, 150, 380, 360, 160, 32, 1082, 2940, 3120, 1680, 480, 64, 9366, 26908, 31080, 19040, 6720, 1344, 128, 94586, 284508, 351344, 236880, 96320, 24192, 3584, 256

Offset: 1

Peter Bala, Sep 13 2011

Define a polynomial sequence P_n(x) by means of the recursion
P_(n+1)(x) = x*(P_n(x)+ P_n(x+1)), with P_0(x) = 1.
The first few polynomials are
P_1(x) = 2*x, P_2(x) = 2*x*(2*x + 1),
P_3(x) = 2*x*(4*x^2 + 6*x + 3), P_4(x) = 2*x*(8*x^3+24*x^2+30*x+13).
The present table shows the coefficients of these polynomials (excluding P_0(x)) in ascending powers of x. The P_n(x) are a polynomial sequence of binomial type. In particular, if we denote P_n(x) by x^[n] then we have the analog of the binomial expansion
(x+y)^[n] = Sum_{k = 0..n} binomial(n,k)*x^[n-k]*y^[k].
There are further analogies between the x^[n] and the monomials x^n.
1) Dobinski-type formula
exp(-x)*Sum_{k >= 0} (-k)^[n]*x^k/k! = (-1)^n*Bell(n,2*x),
where the Bell (or exponential) polynomials are defined as
Bell(n,x) := Sum_{k = 1..n} Stirling2(n,k)*x^k.
Equivalently, the connection constants associated with the polynomial sequences {x^[n]} and {x^n} are (up to signs) the same as the connection constants associated with the polynomial sequences {Bell(n,2*x)} and {Bell(n,x)}. For example, the list of coefficients of x^[4] is [26,60,48,16] and a calculation gives
Bell(4,2*x) = -26*Bell(1,x) + 60*Bell(2,x) - 48*Bell(3,x) + 16*Bell(4,x).
2) Analog of Bernoulli's summation formula
Bernoulli's formula for the sum of the p-th powers of the first n positive integers is
Sum_{k = 1..n} k^p = (1/(p+1))*Sum_{k = 0..p} (-1)^k * binomial(p+1,k)*B_k*n^(p+1-k), where B_k = [1,-1/2,1/6,0,-1/30,...] is the sequence of Bernoulli numbers.
This generalizes to
2*Sum_{k = 1..n} k^[p] = 1/(p+1)*Sum_{k = 0..p} (-1)^k * binomial(p+1,k)*B_k*n^[p+1-k].
The polynomials P_n(x) belong to a family of polynomial sequences P_n(x,t) of binomial type, dependent on a parameter t, and defined recursively by P_(n+1)(x,t)= x*(P_n(x,t)+ t*P_n(x+1,t)), with P_0(x,t) = 1. When t = 0 we have P_n(x,0) = x^n, the monomial polynomials. The present table is the case t = 1. The case t = -2 is (up to signs) A079641. See also A195205 (case t = 2).
Triangle T(n,k) (1 <= k <= n), read by rows, given by (0, 1, 2, 2, 4, 3, 6, 4, 8, 5, 10, ...) DELTA (2, 0, 2, 0, 2, 0, 2, 0, 2, 0, ...) where DELTA is the operator defined in A084938. - Philippe Deléham, Dec 22 2011
T(n,k) is the number of binary relations R on [n] with index = 1 containing exactly k strongly connected components (SCC's) and satisfying the condition that if (x,y) is in R then x and y are in the same SCC. - Geoffrey Critzer, Jan 17 2024

			Triangle begins
n\k|....1......2......3......4......5......6......7
===================================================
..1|....2
..2|....2......4
..3|....6.....12......8
..4|...26.....60.....48.....16
..5|..150....380....360....160.....32
..6|.1082...2940...3120...1680....480.....64
..7|.9366..26908..31080..19040...6720...1344....128
...
Relation with rising factorials for row 4:
x^[4] = 16*x^4+48*x^3+60*x^2+26*x = 2^4*x*(x+1)*(x+2)*(x+3)-6*2^3*x*(x+1)*(x+2)+7*2^2*x*(x+1)-2*x, where [1,7,6,1] is the fourth row of the triangle of Stirling numbers of the second kind A008277.
Generalized Dobinski formula for row 4:
exp(-x)*Sum_{k >= 1} (-k)^[4]*x^k/k! = exp(-x)*Sum_{k >= 1} (16*k^4-48*k^3+60*k^2-26*k)*x^k/k! = 16*x^4+48*x^3+28*x^2+2*x = Bell(4,2*x).
Example of generalized Bernoulli summation formula:
2*(1^[2]+2^[2]+...+n^[2]) = 1/3*(B_0*n^[3]-3*B_1*n^[2]+3*B_2*n^[1]) =
n*(n+1)*(4*n+5)/3, where B_0 = 1, B_1 = -1/2, B_2 = 1/6 are Bernoulli numbers.
From _Philippe Deléham_, Dec 22 2011: (Start)
Triangle (0, 1, 2, 2, 4, 3, 6, ...) DELTA (2, 0, 2, 0, 2, ...) begins:
  1;
  0,    2;
  0,    2,     4;
  0,    6,    12,     8;
  0,   26,    60,    48,    16;
  0,  150,   380,   360,   160,   32;
  0, 1082,  2940,  3120,  1680,  480,   64;
  0, 9366, 26908, 31080, 19040, 6720, 1344, 128;
  ... (End)

Cf. A000629 (row sums), A000670 (one half row sums), A014307 (row polys. at x = 1/2), A079641, A195205, A209849.

# The function BellMatrix is defined in A264428.
# Adds (1,0,0,0, ..) as column 0.
BellMatrix(n -> (-1)^(n+1)*polylog(-n, 2), 10); # Peter Luschny, Jan 29 2016

BellMatrix[f_Function, len_] := With[{t = Array[f, len, 0]}, Table[BellY[n, k, t], {n, 0, len - 1}, {k, 0, len - 1}]];
rows = 12;
M = BellMatrix[(-1)^(#+1) PolyLog[-#, 2]&, rows];
Table[M[[n, k]], {n, 2, rows}, {k, 2, n}] // Flatten (* Jean-François Alcover, Jun 24 2018, after Peter Luschny *)

E.g.f.: F(x,z) := (exp(z)/(2-exp(z)))^x = Sum_{n>=0} P_n(x)*z^n/n!
= 1 + 2*x*z + (2*x+4*x^2)*z^2/2! + (6*x+12*x^2+8*x^3)*z^3/3! + ....
The generating function F(x,z) satisfies the partial differential equation d/dz(F(x,z)) = x*F(x,z) + x*F(x+1,z) and hence the row polynomials P_n(x) satisfy the recurrence relation
P_(n+1)(x)= x*(P_n(x) + P_n(x+1)), with P_0(x) = 1.
In what follows we change notation and write x^[n] for P_n(x).
Relation with the factorial polynomials:
For n >= 1,
x^[n] = Sum_{k = 1..n} (-1)^(n-k)*Stirling2(n,k)*2^k*x^(k),
and its inverse formula
2^n*x^(n) = Sum_{k = 1..n} |Stirling1(n,k)|*x^[k],
where x^(n) denotes the rising factorial x*(x+1)*...*(x+n-1).
Relation with the Bell polynomials:
The alternating n-th row entries (-1)^(n+k)*T(n,k) are the connection coefficients expressing the polynomial Bell(n,2*x) as a linear combination of Bell(k,x), 1 <= k <= n.
The delta operator:
The sequence of row polynomials is of binomial type. If D denotes the derivative operator d/dx then the delta operator D* for this sequence of binomial type polynomials is given by
D* = D/2 - log(cosh(D/2)) = log(2*exp(D)/(exp(D)+1))
= (D/2) - (D/2)^2/2! + 2*(D/2)^4/4! - 16*(D/2)^6/6! + 272*(D/2)^8/8! - ...,
where [1,2,16,272,...] is the sequence of tangent numbers A000182.
D* is the lowering operator for the row polynomials
(D*)x^[n] = n*x^[n-1].
Associated Bernoulli polynomials:
Generalized Bernoulli polynomial GB(n,x) associated with the polynomials x^[n] may be defined by
GB(n,x) := ((D*)/(exp(D)-1))x^[n].
They satisfy the difference equation
GB(n,x+1) - GB(n,x) = n*x^[n-1]
and have the expansion
GB(n,x) = -(1/2)*n*x^[n-1] + (1/2)*Sum_{k = 0..n} binomial(n,k) * B_k * x^[n-k], where B_k denotes the ordinary Bernoulli numbers.
The first few polynomials are
GB(0,x) = 1/2, GB(1,x) = x-3/4, GB(2,x) = 2*x^2-2*x+1/12,
GB(3,x) = 4*x^3-3*x^2-x, GB(4,x) = 8*x^4-4*x^2-4*x-1/60.
It can be shown that
1/(n+1)*(d/dx)(GB(n+1,x)) = Sum_{i = 0..n} 1/(i+1) * Sum_{k = 0..i} (-1)^k *binomial(i,k)*(x+k)^[n].
This generalizes a well-known formula for Bernoulli polynomials.
Relations with other sequences:
Row sums: A000629(n) = 2*A000670(n). Column 1: 2*A000670(n-1). Row polynomials evaluated at x = 1/2: {P_n(1/2)}n>=0 = [1,1,2,7,35,226,...] = A014307.
T(n,k) = A184962(n,k)*2^k. - Philippe Deléham, Feb 17 2013
Also the Bell transform of A076726. For the definition of the Bell transform see A264428. - Peter Luschny, Jan 29 2016
Conjecture: o.g.f. as a continued fraction of Stieltjes type: 1/(1 - 2*x*z/(1 - z/(1 - 2*(x + 1)*z/(1 - 2*z/(1 - 2*(x + 2)*z/(1 - 3*z/(1 - 2*(x + 3)*z/(1 - 4*z/(1 - ... ))))))))). - Peter Bala, Dec 12 2024

a(1) added by Philippe Deléham, Dec 22 2011

A185285 Triangle T(n,k), read by rows, given by (0, 2, 3, 4, 6, 6, 9, 8, 12, 10, 15, ...) DELTA (1, 0, 1, 0, 1, 0, 1, 0, 1, 0, ...) where DELTA is the operator defined in A084938.

Original entry on oeis.org

1, 0, 1, 0, 2, 1, 0, 10, 6, 1, 0, 74, 52, 12, 1, 0, 730, 570, 160, 20, 1, 0, 9002, 7600, 2430, 380, 30, 1, 0, 133210, 119574, 42070, 7630, 770, 42, 1, 0, 2299754, 2170252, 822696, 166320, 19740, 1400, 56, 1, 0, 45375130, 44657106, 17985268, 3956568, 528780, 44604, 2352, 72, 1

Offset: 0

Philippe Deléham, Dec 22 2011

The Bell transform of A004123(n+1). For the definition of the Bell transform see A264428. - Peter Luschny, Jan 18 2016

			Triangle begins :
1
0, 1
0, 2, 1
0, 10, 6, 1
0, 74, 52, 12, 1
0, 730, 570, 160, 20, 1
0, 9002, 7600, 2430, 380, 30, 1
0, 133210, 119574, 42070, 7630, 770, 42, 1

Row sums are A136727.

Cf. A000007, A004123, A000012, A002378, A195205.

(* The function BellMatrix is defined in A264428. *)
a4123[n_] := If[n == 1, 1, PolyLog[-n+1, 2/3]/3];
rows = 10;
M = BellMatrix[a4123[#+1]&, rows];
Table[M[[n, k]], {n, 1, rows}, {k, 1, n}] // Flatten (* Jean-François Alcover, Jun 25 2019 *)

# uses[bell_matrix from A264428]
bell_matrix(lambda n: A004123(n+1), 10) # Peter Luschny, Jan 18 2016

More terms from Jean-François Alcover, Jun 25 2019

cp's OEIS Frontend

A079641 Matrix product of Stirling2-triangle A008277(n,k) and unsigned Stirling1-triangle |A008275(n,k)|.

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A179929 a(n) = 2^n*A(n, -1/2), A(n, x) the Eulerian polynomials.

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A195204 Triangle of coefficients of a sequence of binomial type polynomials.

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A185285 Triangle T(n,k), read by rows, given by (0, 2, 3, 4, 6, 6, 9, 8, 12, 10, 15, ...) DELTA (1, 0, 1, 0, 1, 0, 1, 0, 1, 0, ...) where DELTA is the operator defined in A084938.

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