A049444 -id:A049444 - cp's OEIS frontend

A094646 Generalized Stirling number triangle of first kind.

1, -2, 1, 2, -3, 1, 0, 2, -3, 1, 0, 2, -1, -2, 1, 0, 4, 0, -5, 0, 1, 0, 12, 4, -15, -5, 3, 1, 0, 48, 28, -56, -35, 7, 7, 1, 0, 240, 188, -252, -231, 0, 42, 12, 1, 0, 1440, 1368, -1324, -1638, -231, 252, 114, 18, 1, 0, 10080, 11016, -7900, -12790, -3255, 1533, 1050, 240, 25, 1

Offset: 0

Vladeta Jovovic, May 17 2004

Triangle T(n,k), 0 <= k <= n, read by rows, given by [ -2, 1, -1, 2, 0, 3, 1, 4, 2, 5, ...] DELTA [1, 0, 1, 0, 1, 0, 1, 0, 1, 0, ...] where DELTA is the operator defined in A084938. - Philippe Deléham, Aug 23 2006
From Wolfdieter Lang, Jun 23 2011: (Start)
The row polynomials s(n,x):=Sum_{k=0..n} T(n,k)*x^k satisfy risefac(x-2,n)=s(n,x), with the rising factorials risefac(x-2,n):=Product_{j=0..n-1} (x-2+j), n >= 1, risefac(x-2,0)=1. Compare this with the formula risefac(x,n)=|S1|(n,x), with the row polynomials |S1|(n,x) of A132393 (unsigned Stirling1).
This is the third triangle of an a-family of Sheffer arrays, call them |S1|(a), with e.g.f. of the row polynomials |S1|(a;x;z) = ((1-z)^a)*exp(-x*log(1-z)). In the notation showing the column e.g.f.s this is Sheffer ((1-z)^a,-log(1-z)). In the umbral notation (see the Roman reference, given under A094645) this is called Sheffer for (exp(a*t),1-exp(-t)). For a=0 this becomes the unsigned Stirling1 triangle |S1|(0) = A132393 with row polynomials |S1|(0;n,x) =: s1(n,x).
E.g.f. column number k (with leading zeros): ((1-x)^a)*((-log(1-x))^k)/k!, k >= 0.
E.g.f. for row sums is (1-x)^(a-1), and the e.g.f. for the alternating row sums is (1-x)^(a+1).
Row polynomial recurrence:
  |S1|(a;n,x)=(x+(n-1-a))*|S1|(a;n-1,m), |S1|(a;0,x)=1.
Meixner identity (see the reference under A060338):
  |S1|(a;n,x) - |S1|(a;n,x-1) = n*|S1|(a;n-1,x), n >= 1,
Also (from the corollary 3.7.2 on p. 50 of the Roman reference): |S1|(a;n,x) = (x-a)*|S1|(a;n-1,x+1), n >= 1.
Recurrence: |S1|(a;n,k) = |S1|(a;n-1,k-1) + (n-(a+1))*|S1|(a;n-1,k); |S1|(a;n,k)=0 if n < m, |S1|(a;n,-1)=0, |S1|(a;0,0)=1.
Connection to |Stirling1|=|S1|(0):
  |S1|(a;n,k) = Sum_{p=0..a} |S1|(a;a,p)*abs(Stirling1(n-a,k-p)), n >= a.
The exponential convolution identity is
  |S1|(a;n,x+y) = Sum_{k=0..n} binomial(n,k)*|S1|(a;k,y)*s1(n-k,x), n >= 0, with symmetry x <-> y.
The Sheffer a- and z-sequences are (see the W. Lang link under A006232): Sha(a;n)=A164555(n)/A027642(n) (independent of a) with e.g.f. x/(1-exp(-x)), and the z-sequence has e.g.f. (exp(a*x)-1)/(exp(-x)-1).
The inverse Sheffer matrix has e.g.f. exp(a*z)*exp(x*(1-exp(-z))), in short notation (exp(a*z),1-exp(-z)),
  (or in umbral notation ((1-t)^a,-log(1-t))).
(End)

			Triangle begins
   1;
  -2,  1;
   2, -3,  1;
   0,  2, -3,  1;
   0,  2, -1, -2,  1;
   0,  4,  0, -5,  0,  1;
   ...
risefac(x-2,3) = (x-2)*(x-1)*x = 2*x-3*x^2+x^3.
-1 = T(4,2) = T(3,1) + 1*T(3,2) =  2 + (-3).
T(4,3) = 2*abs(S1(2,3)) - 3*abs(S1(2,2)) + 1*abs(S1(2,1)) = 2*0 - 3*1 + 1*1 = -2.

Cf. A049444, A049458, A094645.

A094646_row := n -> seq((-1)^(n-k)*coeff(expand(pochhammer(x-n+3, n)), x, k), k=0..n): seq(print(A094646_row(n)), n = 0..6); # Peter Luschny, May 16 2013

Flatten[ Table[ CoefficientList[ Pochhammer[x-2, n], x], {n, 0, 10}]] (* Jean-François Alcover, Sep 26 2011 *)

E.g.f.: (1-y)^(2-x).
Sum_{k=0..n} T(n,k)*x^k = A000007(n), A000142(n), A000142(n+1), A001710(n+2), A001715(n+3), A001720(n+4), A001725(n+5), A001730(n+6), A049388(n), A049389(n), A049398(n), A051431(n) for x = 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13 respectively. - Philippe Deléham, Nov 13 2007
If we define f(n,i,a) = Sum_{k=0..n-i} binomial(n,k)*Stirling1(n-k,i)*Product_{j=0..k-1} (-a-j), then |T(n,i)| = |f(n,i,-2)|, for n=1,2,...; i=0..n. - Milan Janjic, Dec 21 2008
From Wolfdieter Lang, Jun 23 2011: (Start)
risefac(x-2,n) = Sum_{k=0..n} T(n,k)*x^k, n >= 0, with the rising factorials (see a comment above).
Recurrence: T(n,k) = T(n-1,k-1) + (n-3)*T(n-1,k); T(n,k)=0 if n < m, T(n,-1)=0, T(0,0)=1.
T(n,k) = 2*abs(S1(n-2,k)) - 3*abs(S1(n-2,k-1)) + abs(S1(n-2,k-2)), n >= 2, with S1(n,k) = Stirling1(n,k) = A048994(n,k).
E.g.f. column number k (with leading zeros):
((1-x)^2)*((-log(1-x))^k)/k!, k >= 0.
E.g.f. for row sums is 1-x, i.e., [1,-1,0,0,...],
  and the e.g.f. for the alternating row sums is (1-x)^3. i.e., [1,-3,3,1,0,0,...]. (End)

A136124 Triangle read by rows: T(n,k) = (-1)^(n+k)*Sum_{j=1..k} s(n,j), where s(n,j) are the signed Stirling numbers of the first kind (n >= 2; 1 <= k <= n-1; s(n,j) = A008275(n,j)).

Original entry on oeis.org

1, 2, 1, 6, 5, 1, 24, 26, 9, 1, 120, 154, 71, 14, 1, 720, 1044, 580, 155, 20, 1, 5040, 8028, 5104, 1665, 295, 27, 1, 40320, 69264, 48860, 18424, 4025, 511, 35, 1, 362880, 663696, 509004, 214676, 54649, 8624, 826, 44, 1, 3628800, 6999840, 5753736, 2655764

Offset: 2

Emeric Deutsch, Dec 23 2007

Sum of entries in row n = n!/2 = A001710(n). T(n,1) = (n-1)! = A000142(n-1). Columns 2,3,4 and 5 yield A001705,A001706,A001707 and A001708, respectively.
See A143491 for the interpretation of these numbers as restricted Stirling numbers of the first kind. See A049444 for a signed version of this array. - Peter Bala, Aug 25 2008
With offset n=0, k=0: triangle T(n,k), read by rows, given by [2,1,3,2,4,3,5,4,6,5,...] DELTA [1,0,1,0,1,0,1,0,1,0,1,0,...] where DELTA is the operator defined in A084938. - Philippe Deléham, Sep 29 2011
With offset n=0, k=0: T(n,k) is the number of ways to seat n people at any number of round tables and serve exactly k of the tables water, some number of the remaining tables red wine, and the rest of the tables white wine. - Geoffrey Critzer, Mar 13 2015

			T(6,3)=71 because (-1)^9*[s(6,1)+s(6,2)+s(6,3)]=-(-120+274-225)=71.
Triangle starts:
    1;
    2,   1;
    6,   5,   1;
   24,  26,   9,   1;
  120, 154,  71,  14,   1;

G. C. Greubel, Table of n, a(n) for the first 50 rows, flattened
Olivier Bodini, Antoine Genitrini, Mehdi Naima, Ranked Schröder Trees, arXiv:1808.08376 [cs.DS], 2018.

Cf. A000142, A008275, A001705, A001706, A001707, A001708, A001710.

Cf. A049444, A143491. [Peter Bala, Aug 25 2008]

A136124_row := proc(n) local k,j; `if`(n=0,1,seq((-1)^(n+1-k)*add(stirling1(n+1,j), j=1..k),k=1..n)) end: seq(print(A136124_row(r)),r=1..6); # Peter Luschny, Sep 29 2011
with(combinat): T:=proc(n, k) options operator, arrow: (-1)^(n+k)*(sum(stirling1(n,j),j=1..k)) end proc: for n from 2 to 11 do seq(T(n,k),k=1..n-1) end do; # yields sequence in triangular form

nn = 10; Map[Select[#, # > 0 &] &,Range[0,nn]!CoefficientList[Series[Exp[(2 + y) Log[1/(1 - x)]], {x, 0, nn}], {x,y}]] // Flatten (* Geoffrey Critzer, Mar 13 2015 *)

E.g.f.: Sum[(1/n!)T(n,k)x^n*t^k, k=1..n-1, n>=2]=1/[(1+t)(1-x)^t]-(1+tx)/(1+t). Generating polynomial of row n = t*Product(j+t, j=2..n-1). T(n,k) is the sum of all products of n-k-1 different integers taken from {2,3,...,n-1}. For example, T(6,3) = 2*3 + 2*4 + 2*5 + 3*4 + 3*5 + 4*5 = 71.

A087745 Numbers A001317 repeated.

Original entry on oeis.org

1, 1, 3, 3, 5, 5, 15, 15, 17, 17, 51, 51, 85, 85, 255, 255, 257, 257, 771, 771, 1285, 1285, 3855, 3855, 4369, 4369, 13107, 13107, 21845, 21845, 65535, 65535, 65537, 65537, 196611, 196611, 327685, 327685, 983055, 983055, 1114129, 1114129

Offset: 0

Philippe Deléham, Oct 02 2003

Triangles A049444, A049459, A051338, A051379, A051523 (Mitrinovic's triangles) mod 2 converted to decimal.

Sequence [1, 3, 5, 15, 17, 51, 85, 255, 257, ...] = A001317.

Cf. A001317, A049444, A049459, A051338, A051378, A051523.

def A087745(n): return sum((bool(~(m:=n>>1)&m-k)^1)<>1)+1)) # Chai Wah Wu, May 02 2023

Definition corrected and edited by Omar E. Pol, Dec 24 2008

A109822 Triangle read by rows: T(n,1)=1, T(n,k) = T(n-1,k) + (n-1)T(n-1, k-1) for 1 <= k <= n.

Original entry on oeis.org

1, 1, 2, 1, 4, 6, 1, 7, 18, 24, 1, 11, 46, 96, 120, 1, 16, 101, 326, 600, 720, 1, 22, 197, 932, 2556, 4320, 5040, 1, 29, 351, 2311, 9080, 22212, 35280, 40320, 1, 37, 583, 5119, 27568, 94852, 212976, 322560, 362880, 1, 46, 916, 10366, 73639, 342964, 1066644

Offset: 1

Emeric Deutsch, Jul 03 2005

T(n,n) = n!. Sum of row n is the signless Stirling number of the first kind s(n,2)(A000254). T(n,k) = A096747(n,k) for 1 <= k <= n.

			T(5,3) = 46 because 18 + 4*7 = 46.
Triangle begins:
Row 1:                    1
Row 2:                 1     2
Row 3:              1     4     6
Row 4:           1     7    18    24
Row 5:        1    11    46    96   120
Row 6:     1    16   101   326   600   720
Row 7:  1    22   197   932  2556  4320  5040

Cf. A000254, A049444, A059518, A096747, A137650.

with(combinat): T:=(n,k)->add(abs(stirling1(n,n-i)),i=0..k-1): for n from 1 to 11 do seq(T(n,k),k=1..n) od; # yields sequence in triangular form T:=proc(n,k) if k=1 then 1 elif k=n then n! else T(n-1,k)+(n-1)*T(n-1,k-1) fi end: for n from 1 to 11 do seq(T(n,k),k=1..n) od; # yields sequence in triangular form. - Emeric Deutsch, Jul 03 2005
A109822_row := proc(n) local k,i;
add(add(abs(combinat[stirling1](n, n-i)), i=0..k)*x^(n-k-1),k=0..n-1);
seq(coeff(%,x,n-k),k=1..n) end:
seq(print(A109822_row(n)),n=1..7); # Peter Luschny, Sep 18 2011

Table[Sum[Abs@ StirlingS1[n, n - i], {i, 0, k - 1}], {n, 10}, {k, n}] // Flatten (* Michael De Vlieger, Aug 17 2017 *)

T(n, k) = Sum_{i=0..k-1} |stirling1(n, n-i)| for 1 <= k <= n.
From Peter Bala, Jul 08 2012: (Start)
E.g.f.: x/(1-x)*{1/(1-x*z)^(1/x) - 1/(1-x*z)} = x*z + (x + 2*x^2)*z^2/2! + (x + 4*x^2 + 6*x^3)*z^3/3! + ... Cf. the e.g.f. of A059518.
(End)

A269952 Triangle read by rows, T(n,k) = Sum_{j=0..n} (-1)^(n-j)C(-j,-n)S2(j,k), S2 the Stirling set numbers A048993, for n>=0 and 0<=k<=n.

Original entry on oeis.org

1, 0, 1, 0, 2, 1, 0, 4, 5, 1, 0, 8, 19, 9, 1, 0, 16, 65, 55, 14, 1, 0, 32, 211, 285, 125, 20, 1, 0, 64, 665, 1351, 910, 245, 27, 1, 0, 128, 2059, 6069, 5901, 2380, 434, 35, 1, 0, 256, 6305, 26335, 35574, 20181, 5418, 714, 44, 1

Offset: 0

Peter Luschny, Apr 10 2016

			1,
0, 1,
0, 2, 1,
0, 4, 5, 1,
0, 8, 19, 9, 1,
0, 16, 65, 55, 14, 1,
0, 32, 211, 285, 125, 20, 1,
0, 64, 665, 1351, 910, 245, 27, 1.

Peter Luschny, Extensions of the binomial

Variant: A143494 (the main entry for this triangle).
A005493 (row sums), A074051 (alt. row sums), A000079 (col. 1), A001047 (col. 2),
A016269 (col. 3), A025211 (col. 4), A000096 (diag. n,n-1), A215862 (diag. n,n-2),
A049444, A136124, A143491 (matrix inverse).
Cf. A048993, A269951.

A269952 := (n,k) -> Stirling2(n+1, k+1) - Stirling2(n, k+1):
seq(seq(A269952(n,k), k=0..n), n=0..9);

Flatten[ Table[ Sum[(-1)^(n-j) Binomial[-j,-n] StirlingS2[j,k], {j,0,n}], {n,0,9}, {k,0,n}]]

T(n, k) = S2(n+1, k+1) - S2(n, k+1).

A360657 Number triangle T associated with 2-Stirling numbers and Lehmer-Comtet numbers (see Comments and Formula section).

Original entry on oeis.org

1, 0, 1, 0, 2, 1, 0, 9, 5, 1, 0, 64, 37, 9, 1, 0, 625, 369, 97, 14, 1, 0, 7776, 4651, 1275, 205, 20, 1, 0, 117649, 70993, 19981, 3410, 380, 27, 1, 0, 2097152, 1273609, 365001, 64701, 7770, 644, 35, 1, 0, 43046721, 26269505, 7628545, 1388310, 174951, 15834, 1022, 44, 1

Offset: 0

Werner Schulte, Feb 15 2023

Triangle T is created using 2-Stirling numbers of the first (A049444) and the second (A143494) kind. The unusual construction is as follows:
  Define A(n, k) by recurrence A(n, k) = A(n-1, k-1) + (k+1) * A(n-1, k) for 0 < k < n with initial values A(n, n) = 1, n >= 0, and A(n, 0) = 0, n > 0. A without column k = 0 is A143494. Let B = A^(-1) matrix inverse of A. B without column k = 0 is A049444. Now define T(m, k) = Sum_{i=0..m-k} B(m-k, i) * A(m-1+i, m-1) for 0 < k <= m = n/2 and T(m, 0) = 0^m for 0 <= m = n/2; T(i, j) = 0 if i < j or j < 0.
Matrix inverse of T is A360753. - Werner Schulte, Feb 21 2023
Conjecture: the transpose of this array is the upper triangular matrix U in the LU factorization of the array of Stirling numbers of the second kind read as a square array; the corresponding lower triangular array L is the triangle of Stirling numbers of the second kind. See the example section below. - Peter Bala, Oct 10 2023

			Triangle T(n, k), 0 <= k <= n, starts:
n\k :  0         1         2        3        4       5      6     7   8  9
==========================================================================
  0 :  1
  1 :  0         1
  2 :  0         2         1
  3 :  0         9         5        1
  4 :  0        64        37        9        1
  5 :  0       625       369       97       14       1
  6 :  0      7776      4651     1275      205      20      1
  7 :  0    117649     70993    19981     3410     380     27     1
  8 :  0   2097152   1273609   365001    64701    7770    644    35   1
  9 :  0  43046721  26269505  7628545  1388310  174951  15834  1022  44  1
  etc.
From _Peter Bala_, Oct 10 2023: (Start)
LU factorization of the square array of Stirling numbers of the second kind (apply Xu, Lemma 2.2):
 / 1               \ / 1   1   1   1  ...\    / 1   1   1    1  ... \
 | 1   1           ||      2   5   9  ...|   |  1   3   6   10  ... |
 | 1   3   1       ||          9  37  ...| = |  1   7  25   65  ... |
 | 1   7   6   1   ||             64  ...|   |  1  15  90  350  ... |
 | ...             ||                 ...|   |  ...                 |
(End)

Wikipedia, LU decomposition
Aimin Xu, Determinants Involving the Numbers of the Stirling-Type, Filomat 33:6 (2019), 1659-1666.

Cf. A049444, A143494, A354794, A354795, A088956, A094587.
Cf. A000007 (column 0), A000169 (column 1), A055869 (column 2).
Cf. A000012 (main diagonal), A000096 (1st subdiagonal), A360753 (matrix inverse).

tabl(m) = {my(n=2*m, A = matid(n), B, T); for( i = 2, n, for( j = 2, i, A[i, j] = A[i-1, j-1] + j * A[i-1, j] ) ); B = A^(-1); T = matrix( m, m, i, j, if( j == 1, 0^(i-1), sum( r = 0, i-j, B[i-j+1, r+1] * A[i-1+r, i-1] ) ) ); }

For the definition of triangle T see Comments section.
Conjectured formulas:
T(n, k) = (Sum_{i=k..n} A354794(n, i) * (i-1)!) / (k-1)! for 0 < k <= n.
T(n, k) - k * T(n, k+1) = A354794(n, k) for 0 <= k <= n.
T(n, 1) = A000169(n) = n^(n-1) for n > 0.
T(n, 2) = A055869(n-1) = n^(n-1) - (n-1)^(n-1) for n > 1.
T(n, k) = (Sum_{i=0..k-1} (-1)^i * binomial(k-1, i) * (n-i)^(n-1)) / (k-1)! for 0 < k <= n.
Sum_{i=1..n} (-1)^(n-i) * binomial(n-1+k, i-1) * T(n, i) * (i-1)! = (k-1)^(n-1) for n > 0 and k >= 0.
Matrix product of A354795 and T without column 0 equals A094587.
Matrix product of T and A354795 without column 0 equals A088956.
E.g.f. of column k > 0: Sum_{n>=k} T(n, k) * t^(n-1) / (n-1)! = (W(-t)/(-t)) * (Sum_{n>=k} A354794(n, k) * t^(n-1) / (n-1)!) where W is the Lambert_W-function.

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A094646 Generalized Stirling number triangle of first kind.

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A136124 Triangle read by rows: T(n,k) = (-1)^(n+k)*Sum_{j=1..k} s(n,j), where s(n,j) are the signed Stirling numbers of the first kind (n >= 2; 1 <= k <= n-1; s(n,j) = A008275(n,j)).

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A087745 Numbers A001317 repeated.

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A109822 Triangle read by rows: T(n,1)=1, T(n,k) = T(n-1,k) + (n-1)T(n-1, k-1) for 1 <= k <= n.

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A269952 Triangle read by rows, T(n,k) = Sum_{j=0..n} (-1)^(n-j)*C(-j,-n)*S2(j,k), S2 the Stirling set numbers A048993, for n>=0 and 0<=k<=n.

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A360657 Number triangle T associated with 2-Stirling numbers and Lehmer-Comtet numbers (see Comments and Formula section).

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A269952 Triangle read by rows, T(n,k) = Sum_{j=0..n} (-1)^(n-j)C(-j,-n)S2(j,k), S2 the Stirling set numbers A048993, for n>=0 and 0<=k<=n.