A195204 -id:A195204 - cp's OEIS frontend

A209849 Triangle read by rows: coefficients of polynomials in Sum_{k = 0..t} k^n * binomial(t,k).

1, 1, 1, 0, 3, 1, -2, 3, 6, 1, 0, -10, 15, 10, 1, 16, -30, -15, 45, 15, 1, 0, 112, -210, 35, 105, 21, 1, -272, 588, 28, -735, 280, 210, 28, 1, 0, -2448, 5292, -2436, -1575, 1008, 378, 36, 1, 7936, -18960, 4140, 20160, -14595, -1575, 2730, 630, 45, 1

Offset: 1

Peter Bala, Mar 15 2012

Repeatedly applying the operator x*d/dx to (1 + x)^t (t a nonnegative integer) and evaluating at x = 1 yields Sum_{k = 0..t} k^n*binomial(t,k) = R(n,t)*2^(t-n), where R(n,t) is a polynomial in t for n = 1,2,.... The polynomial sequence {R(n,t)}_{n>=0} is of binomial type. The first few values are given in the example section below.
This triangle lists the coefficients of these polynomials in ascending powers of t (omitting R(0,t) = 1). A closely related triangle is A102573, which lists the coefficients of the polynomials R(n,t) after factors of t and t*(1 + t) have been removed.
This is the case m = 2 of a family of binomial type polynomials satisfying the recurrence R(n+1,t) = t*(m*(R(n,t) - R(n,t-1)) + R(n,t-1)) with R(0,t) = 1. Case m = 0 gives the falling factorials (A008275); Case m = -1 gives a signed version of A079641.

			Repeatedly applying the operator x*d/dx to (1 + x)^n and evaluating the result at x = 1 yields
Sum_{k = 0..n} k   * binomial(n,k) =  n                * 2^(n-1).
Sum_{k = 0..n} k^2 * binomial(n,k) = (n +   n^2)       * 2^(n-2).
Sum_{k = 0..n} k^3 * binomial(n,k) = (    3*n^2 + n^3) * 2^(n-3).
Triangle begins:
  n\k|    1     2     3     4     5     6     7     8
  = = = = = = = = = = = = = = = = = = = = = = = = = =
  1  |    1
  2  |    1     1
  3  |    0     3     1
  4  |   -2     3     6     1
  5  |    0   -10    15    10     1
  6  |   16   -30   -15    45    15     1
  7  |    0   112  -210    35   105    21     1
  8  | -272   588    28  -735   280   210    28     1
  ...

Seiichi Manyama, Rows n = 1..140, flattened

Columns k=1..3 give A155585(n-1), A383165, A383166.

Cf. A008275, A009006 (alt. row sums), A059419, A063376, A075497, A079641, A102573, A119468, A129062, A154602, A176668, A195204, A383140.

# The function BellMatrix is defined in A264428.
g := n -> 2^n*euler(n,1): BellMatrix(g, 9); # Peter Luschny, Jan 21 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[2^# EulerE[#, 1]&, rows];
Table[M[[n, k]], {n, 2, rows}, {k, 2, n}] // Flatten (* Jean-François Alcover, Jun 24 2018, after Peter Luschny *)

T(n, k) = sum(j=k, n, 2^(n-j)*stirling(n, j, 2)*stirling(j, k, 1)); \\ Seiichi Manyama, Apr 16 2025

# uses[bell_matrix from A264428]
g = lambda n: sum((-2)^(n-k)*factorial(k)*stirling_number2(n,k) for k in (0..n))
bell_matrix(g, 9) # Peter Luschny, Jan 21 2016

def a_row(n):
    s = sum(2^(n-k)*stirling_number2(n, k)*falling_factorial(x, k) for k in (0..n))
    return expand(s).list()[1:]
for n in (1..10): print(a_row(n)) # Seiichi Manyama, Apr 16 2025

T(n,k) = Sum_{j = 0..n} (-1)^(n+k) * (-2)^(n-j) * Stirling2(n,j) * |Stirling1(j,k)|. [corrected by Seiichi Manyama, Apr 16 2025]
E.g.f.: F(x,t) := (1/2 + 1/2*exp(2*x))^t = (1 + tanh(-x))^(-t) = 1 + t*x + (t+t^2)*x^2/2! + (3*t^2+t^3)*x^3/3! + ... satisfies the delay differential equation d/dx(F(x,t)) = 2*F(x,t) - F(x,t-1).
Recurrence for row polynomials R(n,t): R(n+1,t) = t*(2*R(n,t) - R(n,t-1)) with R(0,t) = 1.
Let D be the backward difference operator D(f(x)) = f(x) - f(x-1). Then (x*D)^n(2^x) = 2^(x-n)*R(n,x). Cf. A079641.
Discrete Dobinski-type relation: R(n,x) = 1/2^x*Sum_{k = 0..inf} (2*k)^n*x*(x - 1)*...*(x - k + 1)/k!, valid for x = 0,1,2,.... and n >= 1.
Other Dobinski-type relations: exp(-x)*Sum_{k = 0..inf} R(n,k)*x^k/k! = n-th row polynomial of A075497.
exp(-x)*Sum_{k = 0..inf} R(n,k+1)*x^k/k! = n-th row polynomial of A154602.
i^(-n)*exp(i*x)*Sum_{k = 0..inf} R(n,-k)*(-i*x)^k/k! = n-th row polynomial of A059419 where i = sqrt(-1).
Writing x^[n] in place of R(n,x) we have the analog of the Bernoulli summation formula for powers of integers: Sum_{k = 1..n-1} k^[p] = 1/(p + 1)*Sum_{k = 0..p} 2^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.
n-th row sum R(n,1) equals 2^(n-1). Alternating row sums R(n,-1) starting [-1,0,2,0,-16,0,272,...] are signed tangent numbers - see A009006 and A155585.
R(n+1,2) = 2^n + 4^n = A063376(n).
Triangle as a product of lower triangular arrays equals A075497*A008275.
The triangle of connection constants between the polynomials (x + 1)^[n] and x^[n] appears to be A119468 = (P^2 + 1)/2, where P denotes Pascal's triangle.
Also the Bell transform of the sequence 2^n*E(n,1), E(n,x) the Euler polynomials (A155585). For the definition of the Bell transform see A264428. - Peter Luschny, Jan 21 2016
From Peter Bala, Jun 26 2016: (Start)
With row and column numbering starting at 0:
E.g.f. is exp(x)/cosh(x)*((1 + exp(2*x))/2)^t = 1 + (1 + t)*x + (3*t + t^2)*x^2/2! + (-2 + 3*t + 6*t^2 + t^3)*x^3/3! + ....
Exponential Riordan array [d/dx(f(x)), f(x)] belonging to the Derivative subgroup of the Riordan group, where f(x) = log((1 + exp(2*x))/2) and df/dx = exp(x)/cosh(x) is the e.g.f. for A155585. (End)
T(n,k) = [x^k] Sum_{k=0..n} 2^(n-k) * Stirling2(n,k) * FallingFactorial(x,k). - Seiichi Manyama, Apr 16 2025
E.g.f. of column k (with leading zeros): f(x)^k / k! with f(x) = log(1 + (exp(2*x) - 1)/2). - Seiichi Manyama, Apr 18 2025

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

Original entry on oeis.org

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

A195205 Triangle of coefficients of a sequence of binomial type polynomials.

Original entry on oeis.org

3, 6, 9, 30, 54, 27, 222, 468, 324, 81, 2190, 5130, 4320, 1620, 243, 27006, 68400, 65610, 30780, 7290, 729, 399630, 1076166, 1135890, 618030, 187110, 30618, 2187, 6899262, 19532268, 22212792, 13471920, 4796820, 1020600, 122472, 6561

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) + 2*P_n(x+1)), with P_0(x) = 1.
The first few values are
P_1(x) = 3*x, P_2(x) = 3*x*(3*x + 2),
P_3(x) = 3*x*(9*x^2 + 18*x + 10),
P_4(x) = 3*x*(27*x^3 + 108*x^2 + 156*x + 74).
The present table shows the coefficients of these polynomials (excluding P_0(x)) in ascending powers of x. Compare with A195204.
Triangle T(n,k) (1 <= k <= n), read by rows, given by (0, 2, 3, 4, 6, 6, 9, 8, 12, 10, 15, ...) DELTA (3, 0, 3, 0, 3, 0, 3, 0, 3, ...) where DELTA is the operator defined in A084938. - Philippe Deléham, Dec 22 2011

			Triangle begins
n\k|.....1.......2......3......4......5......6
==============================================
..1|.....3
..2|.....6.......9
..3|....30......54.....27
..4|...222.....468....324.....81
..5|..2190....5130...4320...1620....243
..6|.27006...68400..65610..30780...7290....729
...
Triangle (0, 2, 3, 4, 6, 6, 9, ...) DELTA (3, 0, 3, 0, 3, 0, 3, 0, ...) begins:
1;
0,     3;
0,     6,     9;
0,    30,    54,    27;
0,   222,   468,   324,    81;
0,  2190,  5130,  4320,  1620,   243;
0, 27006, 68400, 65610, 30780,  7290,   729;
... - _Philippe Deléham_, Dec 22 2011

Wikipedia, Binomial type

Cf. A004123 ((1/3)*column 1), A050351 ((1/3)*row sums), A179929 (delta operator coeffs.), A195204.

# The function BellMatrix is defined in A264428.
# Adds (1,0,0,0, ..) as column 0.
BellMatrix(n -> `if`(n=0,3,polylog(-n, 2/3)), 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 = 10;
M = BellMatrix[If[# == 0, 3, PolyLog[-#, 2/3]]&, 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)/(3 - 2*exp(z)))^x = 1 + 3*x*z + (6*x + 9*x^2)*z^2/2! + (30*x + 54*x^2 + 27*x^3)*z^3/3! + ....
The generating function F(x,z) = Sum_{n>=0} P_n(x)*z^n/n! satisfies the partial differential equation d/dz(F(x,z)) = x*F(x,z) + 2*x*F(x+1,z). Hence the row generating polynomials P_n(x) satisfy the recurrence P_(n+1)(x) = x*(P_n(x) + 2*P_n(x+1)), with P_0(x) = 1. The form of the e.g.f. shows that the polynomials P_n(x) are a sequence of binomial type. In what follows we denote P_n(x) by x^[n].
Relation with rising factorials
x^[n] = Sum_{k=1..n} (-1)^(n-k)*Stirling2(n,k)*3^k*x*(x+1)*...*(x+k-1),
and its inverse formula
3^n*x*(x+1)*...*(x+n-1) = Sum_{k=1..n} |Stirling1(n,k)|*x^[k].
The delta operator D*:
The row polynomials form a polynomial sequence of binomial type. If D denotes the derivative operator 1/3*d/dx then the associated delta operator D* is given by D* = D - 2*D^2/2! + 2*D^3/3! + 6*D^4/4! - 30*D^5/5! - ..., where the sequence of coefficients [1, -2, 2, 6, -30, -42, 882, ...] equals (-1)^n*A179929(n). D* is the lowering operator for the row polynomials, that is, (D*)x^[n] = n*x^[n-1].
Generalized Dobinski formula:
exp(-x)*Sum_{k >= 1} (-k)^[n]*x^k/k! = (-1)^n*Bell(n,3*x),
where the Bell (or exponential) polynomials are defined as
Bell(n,x) := Sum_{k = 1..n} Stirling2(n,k)*x^k.
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,3*x) as a linear combination of Bell(k,x), 1 <= k <= n. For example for row 4:
Bell(4,3*x) = -222*Bell(1,x) + 468*Bell(2,x) - 324*Bell(3,x) + 81*Bell(4,x).
Generalized Bernoulli summation formula:
We have the following generalization of Bernoulli's formula for the sum of the powers of integers:
3*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, ...] denotes the sequence of Bernoulli numbers.
Relation with other sequences:
Row sums = 3*A050351(n) for n >= 1. Column 1 = 3*A004123.
T(n,k) = A185285(n,k)*3^k. - Philippe Deléham, Feb 17 2013
Also the Bell transform of 3*A004123. 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 - 3*x*z/(1 - 2*z/(1 - 3*(x + 1)*z/(1 - 4*z/(1 - 3*(x + 2)*z/(1 - 6*z/(1 - 3*(x + 3)*z/(1 - 8*z/(1 - ... ))))))))). - Peter Bala, Dec 12 2024

A184962 Triangle T(n,k), read by rows, given by (0, 1, 2, 2, 4, 3, 6, 4, 8, 5, 10, ...) DELTA (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, 1, 1, 0, 3, 3, 1, 0, 13, 15, 6, 1, 0, 75, 95, 45, 10, 1, 0, 541, 735, 390, 105, 15, 1, 0, 4683, 6727, 3885, 1190, 210, 21, 1, 0, 47293, 71127, 43918, 14805, 3010, 378, 28, 1, 0, 545835

Offset: 0

Philippe Deléham, Dec 22 2011

The Bell transform of the Fubini numbers. For the definition of the Bell transform see A264428. - Peter Luschny, Jan 29 2016

			Triangle begins :
1
0, 1
0, 1, 1
0, 3, 3, 1
0, 13, 15, 6, 1
0, 75, 95, 45, 10, 1

Row sums are A014307(n).

Cf. A000670, A079641, A195204.

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

(* The function BellMatrix is defined in A264428. *)
bm = BellMatrix[(PolyLog[-#, 1/2] + Boole[n == 0])/2 &, 10]; Table[bm[[n, k]], {n, 1, Length[bm]}, {k, 1, n}] // Flatten (* Jean-François Alcover, Mar 31 2016, after Peter Luschny *)

Sum_{k, 0<=k<=n} T(n,k)*x^k = A000007(n), A014307(n), A000629(n) for x = 0, 1, 2 respectively.

cp's OEIS Frontend

A209849 Triangle read by rows: coefficients of polynomials in Sum_{k = 0..t} k^n * binomial(t,k).

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

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A195205 Triangle of coefficients of a sequence of binomial type polynomials.

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A184962 Triangle T(n,k), read by rows, given by (0, 1, 2, 2, 4, 3, 6, 4, 8, 5, 10, ...) DELTA (1, 0, 1, 0, 1, 0, 1, 0, ...) where DELTA is the operator defined in A084938.

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