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

%I A209849 #70 Apr 18 2025 10:59:14
%S A209849 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,
%T A209849 105,21,1,-272,588,28,-735,280,210,28,1,0,-2448,5292,-2436,-1575,1008,
%U A209849 378,36,1,7936,-18960,4140,20160,-14595,-1575,2730,630,45,1
%N A209849 Triangle read by rows: coefficients of polynomials in Sum_{k = 0..t} k^n * binomial(t,k).
%C A209849 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.
%C A209849 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.
%C A209849 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.
%H A209849 Seiichi Manyama, <a href="/A209849/b209849.txt">Rows n = 1..140, flattened</a>
%F A209849 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]
%F A209849 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).
%F A209849 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.
%F A209849 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.
%F A209849 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.
%F A209849 Other Dobinski-type relations: exp(-x)*Sum_{k = 0..inf} R(n,k)*x^k/k! = n-th row polynomial of A075497.
%F A209849 exp(-x)*Sum_{k = 0..inf} R(n,k+1)*x^k/k! = n-th row polynomial of A154602.
%F A209849 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).
%F A209849 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.
%F A209849 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.
%F A209849 R(n+1,2) = 2^n + 4^n = A063376(n).
%F A209849 Triangle as a product of lower triangular arrays equals A075497*A008275.
%F A209849 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.
%F A209849 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
%F A209849 From _Peter Bala_, Jun 26 2016: (Start)
%F A209849 With row and column numbering starting at 0:
%F A209849 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! + ....
%F A209849 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)
%F A209849 T(n,k) = [x^k] Sum_{k=0..n} 2^(n-k) * Stirling2(n,k) * FallingFactorial(x,k). - _Seiichi Manyama_, Apr 16 2025
%F A209849 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
%e A209849 Repeatedly applying the operator x*d/dx to (1 + x)^n and evaluating the result at x = 1 yields
%e A209849 Sum_{k = 0..n} k   * binomial(n,k) =  n                * 2^(n-1).
%e A209849 Sum_{k = 0..n} k^2 * binomial(n,k) = (n +   n^2)       * 2^(n-2).
%e A209849 Sum_{k = 0..n} k^3 * binomial(n,k) = (    3*n^2 + n^3) * 2^(n-3).
%e A209849 Triangle begins:
%e A209849   n\k|    1     2     3     4     5     6     7     8
%e A209849   = = = = = = = = = = = = = = = = = = = = = = = = = =
%e A209849   1  |    1
%e A209849   2  |    1     1
%e A209849   3  |    0     3     1
%e A209849   4  |   -2     3     6     1
%e A209849   5  |    0   -10    15    10     1
%e A209849   6  |   16   -30   -15    45    15     1
%e A209849   7  |    0   112  -210    35   105    21     1
%e A209849   8  | -272   588    28  -735   280   210    28     1
%e A209849   ...
%p A209849 # The function BellMatrix is defined in A264428.
%p A209849 g := n -> 2^n*euler(n,1): BellMatrix(g, 9); # _Peter Luschny_, Jan 21 2016
%t A209849 BellMatrix[f_Function, len_] := With[{t = Array[f, len, 0]}, Table[BellY[n, k, t], {n, 0, len - 1}, {k, 0, len - 1}]];
%t A209849 rows = 12;
%t A209849 M = BellMatrix[2^# EulerE[#, 1]&, rows];
%t A209849 Table[M[[n, k]], {n, 2, rows}, {k, 2, n}] // Flatten (* _Jean-François Alcover_, Jun 24 2018, after _Peter Luschny_ *)
%o A209849 (Sage) # uses[bell_matrix from A264428]
%o A209849 g = lambda n: sum((-2)^(n-k)*factorial(k)*stirling_number2(n,k) for k in (0..n))
%o A209849 bell_matrix(g, 9) # _Peter Luschny_, Jan 21 2016
%o A209849 (Sage)
%o A209849 def a_row(n):
%o A209849     s = sum(2^(n-k)*stirling_number2(n, k)*falling_factorial(x, k) for k in (0..n))
%o A209849     return expand(s).list()[1:]
%o A209849 for n in (1..10): print(a_row(n)) # _Seiichi Manyama_, Apr 16 2025
%o A209849 (PARI) T(n, k) = sum(j=k, n, 2^(n-j)*stirling(n, j, 2)*stirling(j, k, 1)); \\ _Seiichi Manyama_, Apr 16 2025
%Y A209849 Columns k=1..3 give A155585(n-1), A383165, A383166.
%Y A209849 Cf. A008275, A009006 (alt. row sums), A059419, A063376, A075497, A079641, A102573, A119468, A129062, A154602, A176668, A195204, A383140.
%K A209849 sign,easy,tabl
%O A209849 1,5
%A A209849 _Peter Bala_, Mar 15 2012