A167374 -id:A167374 - cp's OEIS frontend

A129184 Shift operator, right.

0, 1, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0

Offset: 1

Gary W. Adamson, Apr 01 2007

Let A129184 = matrix M, then M*V, (V a vector); shifts V to the right, preceded by zeros. Example: M*V, V = [1, 2, 3, ...] = [0, 1, 2, 3, ...]. A129185 = left shift operator.

Given a polynomial sequence p_n(x) with p_0(x)=1 and the lowering and raising operators L and R defined by L P_n(x)= n * P_(n-1)(x) and R P_n(x)= P_(n+1)(x), the matrix T represents the action of R in the p_n(x) basis. For p_n(x) = x^n, L = D = d/dx and R = x. For p_n(x)= x^n/n!, L= DxD and R=D^(-1). - Tom Copeland, Nov 10 2012

			First few rows of the triangle:
  0;
  1, 0;
  0, 1, 0;
  0, 0, 1, 0;
  0, 0, 0, 1, 0;
  ...

Andrew Howroyd, Table of n, a(n) for n = 1..1275

Cf. A129185, A129186.

Cf. A010054.

Infinite lower triangular matrix with all 1's in the subdiagonal and the rest zeros.
From Tom Copeland, Nov 10 2012: (Start)
Let M(t) = I/(I-t*T) = I + t*T + (t*T)^2 + ... where T is the shift operator matrix and I the Identity matrix. Then the inverse matrix is MI(t)=(I-tT) and M(t) is A000012 with each n-th diagonal multiplied by t^n. M(1)=A000012 with inverse MI(1)=A167374. Row sums of M(2), M(3), and M(4) are A000225, A003462, and A002450.
Let E(t)=exp(t*T) with inverse E(-t). Then E(t) is A000012 with each n-th diagonal multiplied by t^n/n! and each row represents e^t truncated at the n+1 term.
The matrix operation b = T*a can be characterized in several ways in terms of the coefficients a(n) and b(n), their o.g.f.s A(x) and B(x), or e.g.f.s EA(x) and EB(x):
  1) b(0) = 0, b(n) = a(n-1),
  2) B(x) = x A(x), or
  3) EB(x) = D^(-1)  EA(x), where D^(-1)x^j/j! = x^(j+1)/(j+1)!.
The operator M(t) can be characterized as
  4)M(t)EA(x)= sum(n>=0)a(n)[e^(x*t)-[1+x*t+...+ (x*t)^(n-1)/(n-1)!]]/t^n
    = exp(a*D_y)[t*e^(x*t)-y*e(x*y)]/(t-y) 
    = [t*e^(x*t)-a*e(x*a)]/(t-a), umbrally where (a)^k=a_k,
  5)[M(t) * a]_n = a(0)t^n +a(1)t^(n-1)+a(2)t^(n-2)+...+a(n).
The exponentiated operator can be characterized as
  6) E(t) A(x) = exp(t*x) A(x),
  7) E(t) EA(x) = exp(t*D^(-1)) EA(x)
  8) [E(t) * a]_n = a(0)t^n/n! + a(1)t^(n-1)/(n-1)! + ... + a(n).
  (End)
a(n) = A010054(n+1). - Andrew Howroyd, Feb 02 2020

Terms a(46) and beyond from Andrew Howroyd, Feb 02 2020

A115717 A divide-and-conquer triangle related to A007583.

Original entry on oeis.org

1, 0, 1, 3, -1, 1, 0, 0, 0, 1, 0, 4, -1, -1, 1, 0, 0, 0, 0, 0, 1, 12, -4, 4, 0, -1, -1, 1, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 4, 0, 0, -1, -1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 16, -4, -4, 4, 0, 0, 0, -1, -1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 4, 0, 0, 0, 0, -1, -1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 48, -16, 16, 0, -4, -4, 4, 0, 0, 0, 0, 0, -1, -1, 1

Offset: 0

Paul Barry, Jan 29 2006

Product of (1-x, x), which is A167374, and number triangle A115715.

			Triangle begins
   1;
   0,   1;
   3,  -1,  1;
   0,   0,  0,  1;
   0,   4, -1, -1,  1;
   0,   0,  0,  0,  0,  1;
  12,  -4,  4,  0, -1, -1,  1;
   0,   0,  0,  0,  0,  0,  0,  1;
   0,   0,  0,  4,  0,  0, -1, -1,  1;
   0,   0,  0,  0,  0,  0,  0,  0,  0,  1;
   0,  16, -4, -4,  4,  0,  0,  0, -1, -1,  1;
   0,   0,  0,  0,  0,  0,  0,  0,  0,  0,  0,  1;
   0,   0,  0,  0,  0,  4,  0,  0,  0,  0, -1, -1,  1;
   0,   0,  0,  0,  0,  0,  0,  0,  0,  0,  0,  0,  0,  1;
  48, -16, 16,  0, -4, -4,  4,  0,  0,  0,  0,  0, -1, -1,  1;

G. C. Greubel, Rows n = 0..50 of the triangle, flattened

Cf. A007583, A115715, A115716 (row sums), A167374.

A115717 := proc(n,k)
    add( A167374(n,j)*A115715(j,k),j=k..n) ;
end proc: # R. J. Mathar, Sep 07 2016

A167374[n_, k_]:= If[k>n-2, (-1)^(n-k), 0];
g[n_, k_]:= g[n, k]= If[k==n, 1, If[k==n-1, -Mod[n, 2], If[n==2*k+2, -4, 0]]]; (* g = A115713 *)
f[n_, k_]:= f[n, k]= If[k==n, 1, -Sum[f[n,j]*g[j,k], {j,k+1,n}]]; (* f=A115715 *)
A115717[n_, k_]:= A115717[n, k]= Sum[A167374[n,j]*f[j,k], {j,k,n}];
Table[A115717[n, k], {n,0,15}, {k,0,n}]//Flatten (* G. C. Greubel, Nov 23 2021 *)

@cached_function
def A115717(n,k):
    def A167374(n, k):
        if (k>n-2): return (-1)^(n-k)
        else: return 0
    def A115713(n,k):
        if (k==n): return 1
        elif (k==n-1): return -(n%2)
        elif (n==2*k+2): return -4
        else: return 0
    def A115715(n,k):
        if (k==0): return 4^(floor(log(n+2, 2)) -1)
        elif (k==n): return 1
        elif (k==n-1): return (n%2)
        else: return (-1)*sum( A115715(n,j+k+1)*A115713(j+k+1,k) for j in (0..n-k-1) )
    return sum( A167374(n, j+k)*A115715(j+k, k) for j in (0..n-k) )
flatten([[A115717(n,k) for k in (0..n)] for n in (0..15)]) # G. C. Greubel, Nov 23 2021

Sum_{k=0..n} T(n, k) = A115716(n).

T(n ,k) = Sum_{j=k..n} A167374(n, j)*A115715(j, k). - R. J. Mathar, Sep 07 2016

A318259 Generalized Worpitzky numbers W_{m}(n,k) for m = 2, n >= 0 and 0 <= k <= n, triangle read by rows.

Original entry on oeis.org

1, -1, 1, 5, -11, 6, -61, 211, -240, 90, 1385, -6551, 11466, -8820, 2520, -50521, 303271, -719580, 844830, -491400, 113400, 2702765, -19665491, 58998126, -93511440, 82661040, -38669400, 7484400, -199360981, 1704396331, -6187282920, 12372329970, -14727913200, 10443232800, -4086482400, 681080400

Offset: 0

Peter Luschny, Sep 06 2018

The triangle can be seen as a member of a family of generalized Worpitzky numbers A028246. See the cross-references for some other members.

The unsigned numbers have row sums A210657 which points to an interpretation of the unsigned numbers as a refinement of marked Schröder paths (see Josuat-Vergès and Kim).

			[0] [      1]
[1] [     -1,         1]
[2] [      5,       -11,        6]
[3] [    -61,       211,     -240,        90]
[4] [   1385,     -6551,    11466,     -8820,     2520]
[5] [ -50521,    303271,  -719580,    844830,  -491400,    113400]
[6] [2702765, -19665491, 58998126, -93511440, 82661040, -38669400, 7484400]

Matthieu Josuat-Vergès and Jang Soo Kim, Touchard-Riordan formulas, T-fractions, and Jacobi's triple product identity, arXiv:1101.5608 [math.CO], 2011.

Row sums are A000007, alternating row sums are A210657.
Cf. T(n,n) = A000680, T(n, 0) = A028296(n) (Gudermannian), A000364 (Euler secant), A241171 (Joffe's differences), A028246 (Worpitzky).
Cf. A167374 (m=0), A028246 & A163626 (m=1), this seq (m=2), A318260 (m=3).

Joffe := proc(n, k) option remember; if k > n then 0 elif k = 0 then k^n else
k*(2*k-1)*Joffe(n-1, k-1)+k^2*Joffe(n-1, k) fi end:
T := (n, k) -> add((-1)^(k-j)*binomial(n-j, n-k)*add((-1)^i*Joffe(n,i)*
binomial(n-i, j), i=0..n), j=0..k):
seq(seq(T(n, k), k=0..n), n=0..6);

Joffe[0, 0] = 1; Joffe[n_, k_] := Joffe[n, k] = If[k>n, 0, If[k == 0,k^n, k*(2*k-1)*Joffe[n-1, k-1] + k^2*Joffe[n-1, k]]];
T[n_, k_] := Sum[(-1)^(k-j)*Binomial[n-j, n-k]*Sum[(-1)^i*Joffe[n, i]* Binomial[n-i, j], {i, 0, n}], {j, 0, k}];
Table[T[n, k], {n, 0, 7}, {k, 0, n}] // Flatten (* Jean-François Alcover, Feb 18 2019, from Maple *)

def EW(m, n):
    @cached_function
    def S(m, n):
        R. = ZZ[]
        if n == 0: return R(1)
        return R(sum(binomial(m*n, m*k)*S(m, n-k)*x for k in (1..n)))
    s = S(m, n).list()
    c = lambda k: sum((-1)^(k-j)*binomial(n-j,n-k)*
        sum((-1)^i*s[i]*binomial(n-i,j) for i in (0..n)) for j in (0..k))
    return [c(k) for k in (0..n)]
def A318259row(n): return EW(2, n)
flatten([A318259row(n) for n in (0..6)])

Let S(n, k) denote Joffe's central differences of zero (A241171) extended to the case n = 0 and k = 0 by prepending a column 1, 0, 0, 0,... to the triangle, then:

T(n,k) = Sum_{j=0..k}((-1)^(k-j)*C(n-j,n-k)*Sum_{i=0..n}((-1)^i*S(n,i)*C(n-i,j))).

A318260 Generalized Worpitzky numbers W_{m}(n,k) for m = 3, n >= 0 and 0 <= k <= n, triangle read by rows.

Original entry on oeis.org

1, -1, 1, 19, -39, 20, -1513, 4705, -4872, 1680, 315523, -1314807, 2052644, -1422960, 369600, -136085041, 710968441, -1484552160, 1548707160, -807206400, 168168000, 105261234643, -661231439271, 1729495989332, -2410936679424, 1889230062720, -789044256000, 137225088000

Offset: 0

Peter Luschny, Sep 06 2018

The triangle can be seen as a member of a family of generalized Worpitzky numbers A028246. See A318259 and the cross-references for some other members.

			[0] [         1]
[1] [        -1,         1]
[2] [        19,       -39,          20]
[3] [     -1513,      4705,       -4872,       1680]
[4] [    315523,  -1314807,     2052644,   -1422960,     369600]
[5] [-136085041, 710968441, -1484552160, 1548707160, -807206400, 168168000]

Cf. T(n,0) ~ A002115(n) (signed), T(n,n) = A014606.

Cf. A167374 (m=0), A028246 & A163626 (m=1), A318259 (m=2), this seq (m=3).

# uses[EW from A318259]
def A318260row(n): return EW(3, n)
print(flatten([A318260row(n) for n in (0..6)]))

Let P(m,n) = Sum_{k=1..n} binomial(m*n, m*k)*P(m, n-k)*x with P(m,0) = 1

and S(n,k) = [x^k]P(3,n), then T(n,k) = Sum_{j=0..k}((-1)^(k-j)*binomial(n-j, n-k)* Sum_{i=0..n}((-1)^i*S(n,i)*binomial(n-i,j))).

A338291 Matrix inverse of the rascal triangle (A077028), read across rows.

Original entry on oeis.org

1, -1, 1, 1, -2, 1, -1, 3, -3, 1, 2, -6, 7, -4, 1, -6, 18, -21, 13, -5, 1, 24, -72, 84, -52, 21, -6, 1, -120, 360, -420, 260, -105, 31, -7, 1, 720, -2160, 2520, -1560, 630, -186, 43, -8, 1, -5040, 15120, -17640, 10920, -4410, 1302, -301, 57, -9, 1

Offset: 0

Werner Schulte, Oct 20 2020

The columns of this triangle are related to factorial numbers (A000142).

There is a family of triangles T(m;n,k) = 1 + m*k*(n-k) for some fixed integer m (for m >= 0 see A296180, Comments) and 0 <= k <= n. They satisfy the equation T(-m;n,k) = 2 - T(m;n,k). The corresponding matrices inverse M = T^(-1) are given by: M(m;n,n) = 1 for n >= 0, and M(m;n,n-1) = m*(1-n) - 1 for n > 0, and M(m;n,k) = (-1)^(n-k) * m * (m * k*(k+1) + 1) * Product_{i=k+1..n-2} (m*(i+1) - 1) for 0 <= k <= n-2. For special cases of the M(m;n,k) see A338817 (m=-1), and A167374 (m=0), and this triangle (m=1).

			The triangle T(n,k) for 0 <= k <= n starts:
n\k :      0      1       2      3      4     5     6   7   8  9
================================================================
  0 :      1
  1 :     -1      1
  2 :      1     -2       1
  3 :     -1      3      -3      1
  4 :      2     -6       7     -4      1
  5 :     -6     18     -21     13     -5     1
  6 :     24    -72      84    -52     21    -6     1
  7 :   -120    360    -420    260   -105    31    -7   1
  8 :    720  -2160    2520  -1560    630  -186    43  -8   1
  9 :  -5040  15120  -17640  10920  -4410  1302  -301  57  -9  1
etc.

Werner Schulte, Proof of the Generalized Formula

Cf. A000007, A000142, A077028.

for(n=0,10,for(k=0,n,if(k==n,print(" 1"),if(k==n-1,print1(-n,", "),print1((-1)^(n-k)*(k^2+k+1)*(n-2)!/k!,", ")))))

1/matrix(10, 10, n, k, n--; k--; if (n>=k, k*(n-k) + 1)) \\ Michel Marcus, Nov 11 2020

T(n,n) = 1 for n >= 0, and T(n,n-1) = -n for n > 0, and T(n,n-2) = n^2 - 3*n + 3 for n > 1, and T(n,k) = (-1)^(n-k) * (k^2 + k + 1) * (n-2)! / k! for 0 <= k <= n-2.
T(n,k) = (2-n) * T(n-1,k) for 0 <= k < n-2.
T(n,k) = T(k+2,k) * (-1)^(n-k) * (n-2)! / k! for 0 <= k <= n-2.
Row sums are A000007(n) for n >= 0.

A167371 Triangle, read by rows, given by [0,1,-1,0,0,0,0,0,0,0,0,...] DELTA [1,0,-1,1,0,0,0,0,0,0,0,...] where DELTA is the operator defined in A084938.

Original entry on oeis.org

1, 0, 1, 0, 1, 1, 0, 0, 1, 1, 0, 0, 0, 1, 1, 0, 0, 0, 0, 1, 1, 0, 0, 0, 0, 0, 1, 1, 0, 0, 0, 0, 0, 0, 1, 1, 0, 0, 0, 0, 0, 0, 0, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1

Offset: 0

Philippe Deléham, Nov 02 2009

Diagonal sums: A060576.

A167374*A154325 formatted as lower triangular matrix. - Philippe Deléham, Nov 19 2009

			Triangle begins:
  1;
  0, 1;
  0, 1, 1;
  0, 0, 1, 1;
  0, 0, 0, 1, 1;
  0, 0, 0, 0, 1, 1; ...

Cf. A097806, A103451.

Sum_{k=0..n} T(n,k)*x^k = A000007(n), A046698(n+1), A111286(n+1), A027327(n) for x= 0, 1, 2, 3 respectively.
G.f.: (1+x^2*y)/(1-x*y). - Philippe Deléham, Nov 09 2013
T(n,k) = T(n-1,k-1) for n > 2, T(0,0) = T(1,1) = T(2,1) = T(2,2) = 1, T(1,0) = T(2,0) = 0, T(n,k) = 0 if k < 0 or if k > n. - Philippe Deléham, Nov 09 2013

cp's OEIS Frontend

A129184 Shift operator, right.

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A115717 A divide-and-conquer triangle related to A007583.

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A318259 Generalized Worpitzky numbers W_{m}(n,k) for m = 2, n >= 0 and 0 <= k <= n, triangle read by rows.

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A318260 Generalized Worpitzky numbers W_{m}(n,k) for m = 3, n >= 0 and 0 <= k <= n, triangle read by rows.

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A338291 Matrix inverse of the rascal triangle (A077028), read across rows.

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A167371 Triangle, read by rows, given by [0,1,-1,0,0,0,0,0,0,0,0,...] DELTA [1,0,-1,1,0,0,0,0,0,0,0,...] where DELTA is the operator defined in A084938.

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