%I #105 Jul 10 2024 15:54:28

%S 1,0,2,1,-1,3,0,3,-3,4,1,-2,7,-6,5,0,4,-8,14,-10,6,1,-3,13,-21,25,-15,

%T 7,0,5,-15,35,-45,41,-21,8,1,-4,21,-49,81,-85,63,-28,9,0,6,-24,71,

%U -129,167,-147,92,-36,10,1,-5,31,-94,201,-295,315,-238,129,-45,11

%N Triangle read by rows: Coefficients for calculation of the sum of all the finite differences from order zero to order k. Sum_{n=0..k} T(n, k)*b(n) = b(0) + b(1) + ... + b(k) + (b(1) - b(0)) + ... + (b(k) - b(k-1)) + ((b(2) - b(1)) - (b(1) - b(0))) + ... .

%C If we want to calculate the sum of finite differences for a sequence b(n):

%C b(0)*T(0, n) + ... + b(n)*T(n, n) = b(0) + b(1) + ... + b(n) + (b(1) - b(0)) + ... + (b(n) - b(n-1)) + ((b(2) - b(1)) - (b(1) - b(0))) + ... This sum includes the sequence b(n) itself. This defines an invertible linear sequence transformation with a deep connection to Bernoulli numbers and other interesting sequences of rational numbers.

%C From _Thomas Scheuerle_, Apr 29 2024: (Start)

%C These are the coefficients of the polynomials defined by the recurrence: P(k, x) = P(k - 1, x) + (x^2 - x)*P(k - 2, x) + 1, with P(-1, x) = 0 and P(0, x) = 1. This can also be expressed as P(k, x) = Sum_{m=1..k+1} binomial(k+2 - m, m)*(x^2 - x)^(m - 1) = Sum_{n=0..k} T(n, k)*x^(k-n). If we would evaluate P(k, t) as sequence for some fixed t then we get the expansion of 1/((1 - x)*(1+(t-1)*x)*(1 - t*x)).

%C We may replace (x^2 - x) by (x^(-2) - x^(-1)) to get the coefficients in reverse order: x^k*Sum_{m=1..k+1} binomial(k+2 - m, m)*(x^(-2) - x^(-1))^(m - 1) = Sum_{n=0..k} T(n, k)*x^n = F(k, x). If we would evaluate F(k, t) as sequence for some fixed t then we get the expansion of 1/((1 - x)*(1 - (t-1)*x)*(1 - t*x)). (End)

%F b(0)*T(0, m) + b(1)*T(1, m) + ... + b(m)*T(m, m)

%F = Sum_{j=0..m} Sum_{n=0..m-j} Sum_{k=0..n} (-1)^k*binomial(n, k)*b(j+n-k)

%F = Sum_{n=0..m} b(n)*Sum_{j=n..m}(-1)^(j+n)*binomial(j+1, n).

%F T(n, k) = Sum_{m=n..k}(-1)^(m+n)*binomial(m+1, n).

%F T(n, k) = (1/2)*(-1)^n*(2*(-1)^k*binomial(2+k, n)*Hypergeometric2F1(1, k+3, k+3-n, -1)+(-1/2)^n*(2^(n+1) - 1)), where Hypergeometric2F1 is the Gaussian hypergeometric function 2F1 as defined in Mathematica. - _Thomas Scheuerle_, Apr 29 2024

%F T(k, k) = A000027(k+1) The positive integers.

%F |T(k-1, k)| = A000217(k) The triangular numbers.

%F T(k-2, k) = A004006(k).

%F |T(k-3, k)| = A051744(k).

%F T(0, k*2) = 1.

%F T(0, k*2 + 1) = 0.

%F T(1, k*2 + 1) = k + 2.

%F T(1, k*2 + 2) = -(k + 1).

%F T(n, k) with constant n and variable k, a linear recurrence relation with characteristic polynomial (x-1)*(x+1)^(n+1).

%F Sum_{n=0..k} T(n, k)*B_n = 1. B_n is the n-th Bernoulli number with B_1 = 1/2. B_n = A164555(n)/A027642(n).

%F Sum_{n=0..k} T(n, k)*(1 - B_n) = k.

%F Sum_{n=0..k} T(n, k)*(2*n - 3+3*B_n) = k^2.

%F Sum_{n=0..k} T(n, k)*A032346(n) = A032346(k+1).

%F From _Thomas Scheuerle_, Apr 29 2024: (Start)

%F Sum_{n=0..k} T(n, k)*A000110(n+1) = A000110(k+2) - 1.

%F Sum_{n=0..k} T(n, k)*(1/(1+n)) = H(1+floor(k/2)), where H(k) is the harmonic number A001008(k)/A002805(k). (End)

%F Sum_{n=0..k} T(n, k)*c(n) = c(k). C(k) = {-1, 0, 1/2, 1/2, 1/8, -7/20, ...} this sequence of rational numbers can be defined recursively: c(0) = -1, c(m) = (-c(m-1) + Sum_{k=0..m-1} A130595(m+1, k)*c(k))/m.

%F c(m) is an eigensequence of this transformation, all eigensequences are c(m) multiplied by any factor.

%F Sum_{n=0..k} T(n, k)*A000045(n) = 2*(A000045(2*floor((k+1)/2) - 1) - 1). A000045 are the Fibonacci numbers.

%F Sum_{n=0..k} T(n, k)*A000032(n) = A000032(2*floor(k/2)+2) - 2. A000032 are the Lucas numbers.

%F Sum_{n=0..k} T(n, k)*A001045(n) = A145766(floor((k+1)/2)). A001045 is the Jacobsthal sequence.

%F This sequence acting as an operator onto a monomial n^w:

%F Sum_{n=0..k} T(n, k)*n^w = (1/(w+1))*k^(w+1) + Sum_{v=1..w} ((v+B_v)*(w)_v/v!)*k^(w+1-v) - A052875(w) + O_k(w) (w)_v is the falling factorial. If k > w-1 then O_k(w) = 0. If k <= w-1 then O_k(w) is A084416(w, 2+k), the sequence with the exponential generating function: (e^x-1)^(2+k)/(2-e^x).

%F From _Thomas Scheuerle_, Apr 29 2024: (Start)

%F This sequence acting by its inverse operator onto a monomial k^w:

%F Sum_{n=0..k} T(n, k)*( Sum_{m=0..k} ((-1)^(1+m+k)*binomial(k, m)*(2^(k-m) - 1)*n^m + A344037(m)*B_n) ) = k^w - A372245(w, k+3), note that A372245(w, k+3) = 0 if k+3 > w. B_n is the n-th Bernoulli number with B_1 = 1/2.

%F How this sequence will act as an operator onto a Dirichlet series may be developed by the formulas below:

%F Sum_{n=0..k} T(n, k)*2^n = A000295(k+2).

%F Sum_{n=0..k} T(n, k)*3^n = A000392(k+3).

%F Sum_{n=0..k} T(n, k)*4^n = A016208(k).

%F Sum_{n=0..k} T(n, k)*5^n = A016218(k).

%F Sum_{n=0..k} T(n, k)*6^n = A016228(k).

%F Sum_{n=0..k} T(n, k)*7^n = A016241(k).

%F Sum_{n=0..k} T(n, k)*8^n = A016249(k).

%F Sum_{n=0..k} T(n, k)*9^n = A016256(k).

%F Sum_{n=0..k} T(n, k)*10^n = A016261(k).

%F Sum_{n=0..k} T(n, k)*m^n = m^2*m^k/(m-1) - (m-1)^2*(m-1)^k/(m-2) + 1/((m-1)*(m-2)), for m > 2.

%F Sum_{n=0..k} T(n, k)*( m*B_n + (m-1)*Sum_{t=1..m} t^n )*(1/m^2) = m^k, for m > 0. B_n is the n-th Bernoulli number with B_1 = 1/2.

%F Sum_{n=0..k} T(n, k) zeta(-n) = Sum_{j=0..k} (-1)^(1+j)/(2+j) = (-1)^(k+1)*LerchPhi(-1, 1, k+3) - 1 + log(2).

%F Sum_{n=0..k} T(k - n, k)*2^n = A000975(k+1)

%F Sum_{n=0..k} T(k - n, k)*3^n = A091002(k+2)

%F Sum_{n=0..k} T(k - n, k)*4^n = A249997(k). (End)

%e Triangle begins with T(n, k):

%e n= 0, 1, 2, 3, 4, 5, 6, 7, 8

%e k=0 1

%e k=1 0, 2

%e k=2 1, -1, 3

%e k=3 0, 3, -3, 4

%e k=4 1, -2, 7, -6, 5

%e k=5 0, 4, -8, 14, -10, 6

%e k=6 1, -3, 13, -21, 25, -15, 7

%e k=7 0, 5, -15, 35, -45, 41, -21, 8

%e k=8 1, -4, 21, -49, 81, -85, 63, -28, 9

%e ...

%o (PARI) A341091(n, k) = sum(m=n, k,(-1)^(m+n)*binomial(m+1, n))

%o (PARI) A341091(n, k) = (1/2)*(-1)^n*(2*(-1)^k*binomial(2+k, n)*hypergeom([1,k+3],k+3-n,-1)+(-1/2)^n*(2^(n+1)-1)) \\ _Thomas Scheuerle_, Apr 29 2024

%Y Cf. A000027, A000032, A000045, A000110, A000217.

%Y Cf. A001045, A004006, A032346, A051744, A052875.

%Y Cf. A084416, A130595, A145766.

%Y Cf. A027642, A164555 (Numerators and denominators of Bernoulli numbers).

%Y Cf. A001008, A002805 (Numerators and denominators of harmonic numbers).

%Y Cf. A344037, A372245.

%Y Sequences below will be obtained by evaluation of the associated polynomials:

%Y Cf. A000295, A000392, A000975, A016208.

%Y Cf. A016218, A016228, A016241, A016249.

%Y Cf. A016256, A016261, A249997.

%K sign,tabl,easy

%O 0,3

%A _Thomas Scheuerle_, Feb 13 2022