A051338 Generalized Stirling number triangle of first kind.
1, -6, 1, 42, -13, 1, -336, 146, -21, 1, 3024, -1650, 335, -30, 1, -30240, 19524, -5000, 635, -40, 1, 332640, -245004, 74524, -11985, 1075, -51, 1, -3991680, 3272688, -1139292, 218344, -24885, 1687, -63, 1, 51891840, -46536624, 18083484, -3977764, 541849, -46816, 2506, -76, 1
Offset: 0
Examples
{1}; {-6,1}; {42,-13,1}; {-336,146,-21,1}; ... s(2,x)= 42-13*x+x^2; S1(2,x)= -x+x^2 (Stirling1).
References
- Mitrinovic, D. S.; Mitrinovic, R. S.; Tableaux d'une classe de nombres relies aux nombres de Stirling. Univ. Beograd. Pubi. Elektrotehn. Fak. Ser. Mat. Fiz. No. 77 1962, 77 pp.
Links
- Reinhard Zumkeller, Rows n = 0..125 of triangle, flattened
Crossrefs
Programs
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Haskell
a051338 n k = a051338_tabl !! n !! k a051338_row n = a051338_tabl !! n a051338_tabl = map fst $ iterate (\(row, i) -> (zipWith (-) ([0] ++ row) $ map (* i) (row ++ [0]), i + 1)) ([1], 6) -- Reinhard Zumkeller, Mar 11 2014
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Mathematica
t[n_, i_] = Sum[(-1)^k*Binomial[n, k]*Pochhammer[6, k]*StirlingS1[n - k, i], {k, 0, n - i}]; Flatten[Table[t[n, i], {n, 0, 8}, {i, 0, n}]][[1 ;; 45]] (* Jean-François Alcover, Jun 01 2011, after Milan Janjic *)
Formula
a(n, m)= a(n-1, m-1) - (n+5)*a(n-1, m), n >= m >= 0; a(n, m) := 0, n
E.g.f. for m-th column of signed triangle: ((log(1+x))^m)/(m!*(1+x)^6).
Triangle (signed) = [ -6, -1, -7, -2, -8, -3, -9, -4, -10, ...] DELTA A000035; triangle (unsigned) = [6, 1, 7, 2, 8, 3, 9, 4, 10, 5, 11, ...] DELTA A000035; where DELTA is Deléham's operator defined in A084938.
If we define f(n,i,a)=sum(binomial(n,k)*stirling1(n-k,i)*product(-a-j,j=0..k-1),k=0..n-i), then T(n,i) = f(n,i,6), for n=1,2,...;i=0...n. - Milan Janjic, Dec 21 2008
A051339 Generalized Stirling number triangle of first kind.
1, -7, 1, 56, -15, 1, -504, 191, -24, 1, 5040, -2414, 431, -34, 1, -55440, 31594, -7155, 805, -45, 1, 665280, -434568, 117454, -16815, 1345, -57, 1, -8648640, 6314664, -1961470, 336049, -34300, 2086, -70, 1, 121080960, -97053936, 33775244, -6666156, 816249, -63504, 3066, -84, 1
Offset: 0
Comments
a(n,m)= ^7P_n^m in the notation of the given reference with a(0,0) := 1. The monic row polynomials s(n,x) := sum(a(n,m)*x^m,m=0..n) which are s(n,x)= product(x-(7+k),k=0..n-1), n >= 1 and s(0,x)=1 satisfy s(n,x+y) = sum(binomial(n,k)*s(k,x)*S1(n-k,y),k=0..n), with the Stirling1 polynomials S1(n,x)=sum(A008275(n,m)*x^m, m=1..n) and S1(0,x)=1. In the umbral calculus (see the S. Roman reference given in A048854) the s(n,x) polynomials are called Sheffer for (exp(7*t),exp(t)-1).
Examples
{1}; {-7,1}; {56,-15,1}; {-504,191,-24,1}; ... s(2,x)= 56-15*x+x^2; S1(2,x)= -x+x^2 (Stirling1).
Links
- Reinhard Zumkeller, Rows n = 0..125 of triangle, flattened
- D. S. Mitrinovic, M. S. Mitrinovic, Tableaux d'une classe de nombres reliés aux nombres de Stirling, Univ. Beograd. Pubi. Elektrotehn. Fak. Ser. Mat. Fiz. 77 (1962).
Crossrefs
Programs
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Haskell
a051339 n k = a051339_tabl !! n !! k a051339_row n = a051339_tabl !! n a051339_tabl = map fst $ iterate (\(row, i) -> (zipWith (-) ([0] ++ row) $ map (* i) (row ++ [0]), i + 1)) ([1], 7) -- Reinhard Zumkeller, Mar 11 2014
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Mathematica
a[n_, m_] := Pochhammer[m + 1, n - m] SeriesCoefficient[Log[1 + x]^m/(1 + x)^7, {x, 0, n}]; Table[a[n, m], {n, 0, 8}, {m, 0, n}] // Flatten (* Jean-François Alcover, Oct 29 2019 *)
Formula
a(n, m)= a(n-1, m-1) - (n+6)*a(n-1, m), n >= m >= 0; a(n, m) := 0, n
E.g.f. for m-th column of signed triangle: ((log(1+x))^m)/(m!*(1+x)^7).
Triangle (signed) = [ -7, -1, -8, -2, -9, -3, -10, -4, -11, -5, ...] DELTA A000035; triangle (unsigned) = [7, 1, 8, 2, 9, 3, 10, 4, ...] DELTA A000035; where DELTA is Deléham's operator defined in A084938.
If we define f(n,i,a)=sum(binomial(n,k)*stirling1(n-k,i)*product(-a-j,j=0..k-1),k=0..n-i), then T(n,i) = f(n,i,7), for n=1,2,...;i=0...n. [From Milan Janjic, Dec 21 2008]
A051379 Generalized Stirling number triangle of first kind.
1, -8, 1, 72, -17, 1, -720, 242, -27, 1, 7920, -3382, 539, -38, 1, -95040, 48504, -9850, 995, -50, 1, 1235520, -725592, 176554, -22785, 1645, -63, 1, -17297280, 11393808, -3197348, 495544, -45815, 2527, -77, 1, 259459200, -188204400, 59354028, -10630508, 1182769, -83720, 3682, -92, 1
Offset: 0
Comments
a(n,m)= ^8P_n^m in the notation of the given reference with a(0,0) := 1. The monic row polynomials s(n,x) := sum(a(n,m)*x^m,m=0..n) which are s(n,x)= product(x-(8+k),k=0..n-1), n >= 1 and s(0,x)=1 satisfy s(n,x+y) = sum(binomial(n,k)*s(k,x)*S1(n-k,y),k=0..n), with the Stirling1 polynomials S1(n,x)=sum(A008275(n,m)*x^m, m=1..n) and S1(0,x)=1. In the umbral calculus (see the S. Roman reference given in A048854) the s(n,x) polynomials are called Sheffer for (exp(8*t),exp(t)-1).
Examples
{1}; {-8,1}; {72,-17,1}; {-720,242,-27,1}; ... s(2,x)=72-17*x+x^2; S1(2,x)= -x+x^2 (Stirling1).
Links
- Reinhard Zumkeller, Rows n = 0..125 of triangle, flattened
- D. S. Mitrinovic, M. S. Mitrinovic, Tableaux d'une classe de nombres reliés aux nombres de Stirling, Univ. Beograd. Pubi. Elektrotehn. Fak. Ser. Mat. Fiz. 77 (1962).
Crossrefs
Programs
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Haskell
a051379 n k = a051379_tabl !! n !! k a051379_row n = a051379_tabl !! n a051379_tabl = map fst $ iterate (\(row, i) -> (zipWith (-) ([0] ++ row) $ map (* i) (row ++ [0]), i + 1)) ([1], 8) -- Reinhard Zumkeller, Mar 12 2014
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Mathematica
a[n_, m_] := Pochhammer[m + 1, n - m] SeriesCoefficient[Log[1 + x]^m/(1 + x)^8, {x, 0, n}]; Table[a[n, m], {n, 0, 8}, {m, 0, n}] // Flatten (* Jean-François Alcover, Oct 29 2019 *)
Formula
a(n, m)= a(n-1, m-1) - (n+7)*a(n-1, m), n >= m >= 0; a(n, m) := 0, n
E.g.f. for m-th column of signed triangle: ((log(1+x))^m)/(m!*(1+x)^8).
Triangle (signed) = [ -8, -1, -9, -2, -10, -3, -11, -4, -12, ...] DELTA A000035; triangle (unsigned) = [8, 1, 9, 2, 10, 3, 11, 4, 12, 5, ...] DELTA A000035; where DELTA is Deléham's operator defined in A084938.
If we define f(n,i,a)=sum(binomial(n,k)*stirling1(n-k,i)*product(-a-j,j=0..k-1),k=0..n-i), then T(n,i) = f(n,i,8), for n=1,2,...;i=0...n. - Milan Janjic, Dec 21 2008
Extensions
Typo fixed in data by Reinhard Zumkeller, Mar 12 2014
A176733 a(n) = (n+6)*a(n-1) + (n-1)*a(n-2), a(-1)=0, a(0)=1.
1, 7, 57, 527, 5441, 61959, 770713, 10391023, 150869313, 2346167879, 38896509881, 684702346767, 12752503850497, 250514001320647, 5176062576469401, 112204510124346479, 2546140161382663553, 60356495873790805383, 1491840283714484609593, 38382424018590349736719
Offset: 0
Comments
a(n) enumerates the possibilities for distributing n beads, n>=1, labeled differently from 1 to n, over a set of (unordered) necklaces, excluding necklaces with exactly one bead, and k=7 indistinguishable, ordered, fixed cords, each allowed to have any number of beads. Beadless necklaces as well as beadless cords contribute a factor 1 in the counting, e.g., a(0):= 1*1 =1. See A000255 for the description of a fixed cord with beads. This produces for a(n) the exponential (aka binomial) convolution of the subfactorial sequence {A000166(n)} and the sequence {A001730(n+6) = (n+6)!/6!}. See the necklaces and cords problem comment in A000153. Therefore the recurrence with inputs holds. This comment derives from a family of recurrences found by Malin Sjodahl for a combinatorial problem for certain quark and gluon diagrams (Feb 27 2010).
Examples
Necklaces and 7 cords problem. For n=4 one considers the following weak 2-part compositions of 4: (4,0), (3,1), (2,2), and (0,4), where (1,3) does not appear because there are no necklaces with 1 bead. These compositions contribute respectively !4*1,binomial(4,3)*!3*c7(1), (binomial(4,2)*!2)*c7(2), and 1*c7(4) with the subfactorials !n:=A000166(n) (see the necklace comment there) and the c7(n):=A001730(n+6) numbers for the pure 7-cord problem (see the remark on the e.g.f. for the k-cord problem in A000153; here for k=7: 1/(1-x)^7). This adds up as 9 + 4*2*7 + (6*1)*56 + 5040 = 5441 = a(4).
Links
- Robert Israel, Table of n, a(n) for n = 0..442
Crossrefs
Cf. A176732 (necklaces and k=6 cords).
Programs
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Maple
f:= proc(n) option remember; (n+6)*procname(n-1) + (n-1)*procname(n-2) end proc: f(-1):= 0: f(0):= 1: map(f, [$0..30]); # Robert Israel, Dec 01 2024
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Mathematica
Table[(-1)^n HypergeometricPFQ[{8, -n}, {}, 1], {n, 0, 20}] (* Benedict W. J. Irwin, May 29 2016 *)
Formula
E.g.f. (exp(-x)/(1-x))*(1/(1-x)^7) = exp(-x)/(1-x)^8, equivalent to the given recurrence.
a(n) = A086764(n+7,7).
a(n) = (-1)^n*2F0(8,-n;;1). - Benedict W. J. Irwin, May 29 2016
A161742 Third left hand column of the RSEG2 triangle A161739.
1, 4, 13, 30, -14, -504, 736, 44640, -104544, -10644480, 33246720, 5425056000, -20843695872, -5185511654400, 23457840537600, 8506857655296000, -44092609863966720, -22430879475779174400, 130748316971139072000
Offset: 2
Crossrefs
Programs
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Maple
nmax:=21; for n from 0 to nmax do A008955(n,0):=1 end do: for n from 0 to nmax do A008955(n,n):=(n!)^2 end do: for n from 1 to nmax do for m from 1 to n-1 do A008955(n,m):= A008955(n-1,m-1)*n^2+A008955(n-1,m) end do: end do: for n from 1 to nmax do A028246(n,1):=1 od: for n from 1 to nmax do A028246(n,n):=(n-1)! od: for n from 3 to nmax do for m from 2 to n-1 do A028246(n,m):=m*A028246(n-1,m)+(m-1)*A028246(n-1,m-1) od: od: for n from 2 to nmax do a(n):=sum(((-1)^k/((k+1)!*(k+2)!)) *(n!)*A028246(n,k+2)* A008955(k+1,k),k=0..n-2) od: seq(a(n),n=2..nmax);
A161743 Fourth left hand column of the RSEG2 triangle A161739.
1, 10, 73, 425, 1561, -2856, -73520, 380160, 15376416, -117209664, -7506967104, 72162155520, 7045087741056, -80246202992640, -11448278791372800, 149576169325363200, 30017051616972275712, -440857664887810867200
Offset: 3
Crossrefs
Programs
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Maple
nmax:=21; for n from 0 to nmax do A008955(n,0):=1 end do: for n from 0 to nmax do A008955(n,n):=(n!)^2 end do: for n from 1 to nmax do for m from 1 to n-1 do A008955(n,m):= A008955(n-1,m-1)*n^2+A008955(n-1,m) end do: end do: for n from 1 to nmax do A028246(n,1):=1 od: for n from 1 to nmax do A028246(n,n):=(n-1)! od: for n from 3 to nmax do for m from 2 to n-1 do A028246(n,m):=m*A028246(n-1,m)+(m-1)*A028246(n-1,m-1) od: od: for n from 3 to nmax do a(n) := sum(((-1)^k/((k+2)!*(k+3)!))*(n!)*A028246(n,k+3)* A008955(k+2,k), k=0..n-3) od: seq(a(n),n=3..nmax);
A162995 A scaled version of triangle A162990.
1, 3, 1, 12, 4, 1, 60, 20, 5, 1, 360, 120, 30, 6, 1, 2520, 840, 210, 42, 7, 1, 20160, 6720, 1680, 336, 56, 8, 1, 181440, 60480, 15120, 3024, 504, 72, 9, 1, 1814400, 604800, 151200, 30240, 5040, 720, 90, 10, 1
Offset: 1
Comments
We get this scaled version of triangle A162990 by dividing the coefficients in the left hand columns by their 'top-values' and then taking the square root.
T(n,k) = A173333(n+1,k+1), 1 <= k <= n. - Reinhard Zumkeller, Feb 19 2010
T(n,k) = A094587(n+1,k+1), 1 <= k <= n. - Reinhard Zumkeller, Jul 05 2012
Examples
The first few rows of the triangle are: [1] [3, 1] [12, 4, 1] [60, 20, 5, 1]
Links
- Reinhard Zumkeller, Rows n = 1..150 of triangle, flattened
Crossrefs
Programs
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Haskell
a162995 n k = a162995_tabl !! (n-1) !! (k-1) a162995_row n = a162995_tabl !! (n-1) a162995_tabl = map fst $ iterate f ([1], 3) where f (row, i) = (map (* i) row ++ [1], i + 1) -- Reinhard Zumkeller, Jul 04 2012
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Maple
a := proc(n, m): (n+1)!/(m+1)! end: seq(seq(a(n, m), m=1..n), n=1..9); # Johannes W. Meijer, revised Nov 23 2012
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Mathematica
Table[(n+1)!/(m+1)!, {n, 10}, {m, n}] (* Paolo Xausa, Mar 31 2024 *)
Formula
a(n,m) = (n+1)!/(m+1)! for n = 1, 2, 3, ..., and m = 1, 2, ..., n.
A167557 The lower left triangle of the ED1 array A167546.
1, 1, 4, 2, 12, 32, 6, 48, 160, 384, 24, 240, 960, 2688, 6144, 120, 1440, 6720, 21504, 55296, 122880, 720, 10080, 53760, 193536, 552960, 1351680, 2949120, 5040, 80640, 483840, 1935360, 6082560, 16220160, 38338560, 82575360
Offset: 1
Comments
We discovered that the numbers that appear in the lower left triangle of the ED1 array A167546 (m <= n) behave in a regular way, see the formula below. This rather simple regularity doesn't show up in the upper right triangle of the ED1 array (m > n).
Examples
The first few triangle rows are: [1] [1, 4] [2, 12, 32] [6, 48, 160, 384] [24, 240, 960, 2688, 6144] [120, 1440, 6720, 21504, 55296, 122880]
Links
- G. C. Greubel, Table of n, a(n) for the first 50 rows
Crossrefs
Programs
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Maple
a := proc(n, m): 4^(m-1)*(m-1)!*(n+m-2)!/(2*m-2)! end: seq(seq(a(n, m), m=1..n), n=1..8); # Johannes W. Meijer, revised Nov 23 2012
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Mathematica
Flatten[Table[(4^(m-1) (m-1)!(n+m-2)!)/(2m-2)!,{n,10},{m,n}]] (* Harvey P. Dale, Sep 29 2013 *)
Formula
a(n,m) = 4^(m-1)*(m-1)!*(n+m-2)!/(2*m-2)!.
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
Comments
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)
Examples
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.
Programs
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Maple
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
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Mathematica
Flatten[ Table[ CoefficientList[ Pochhammer[x-2, n], x], {n, 0, 10}]] (* Jean-François Alcover, Sep 26 2011 *)
Formula
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)
A134141 Generalized unsigned Stirling1 triangle, S1p(7).
1, 7, 1, 56, 21, 1, 504, 371, 42, 1, 5040, 6440, 1295, 70, 1, 55440, 114520, 36225, 3325, 105, 1, 665280, 2116800, 983920, 135975, 7105, 147, 1, 8648640, 40884480, 26714800, 5199145, 398860, 13426, 196, 1, 121080960, 826338240, 735469280
Offset: 1
Comments
Signed lower triangular matrix (-1)^(n-m)*a(n,m) is inverse to matrix A092082(n, m) =: S2(7; n,m). The monic row polynomials E(n,x) := sum(a(n,m)*x^m, m=1..n), E(0,x) := 1 are exponential convolution polynomials (see A039692 for the definition and a Knuth reference).
a(n,m) enumerates unordered increasing n-vertex m-forests composed of m unary trees (out-degree r from {0,1}) whose vertices of depth (distance from the root) j>=1 come in j+6 colors. The k roots (j=0) each come in one (or no) color. - Wolfdieter Lang, Oct 05 2007
A triangle of numbers related to triangle A132166.
a(n,1)= A001730(n,5), n>=1. a(n,m)=: S1p(7; n,m), a member of a sequence of lower triangular Jabotinsky matrices with nonnegative entries, including S1p(1; n, m)= A008275 (unsigned Stirling first kind), S1p(2; n,m)= A008297(n, m) (unsigned Lah numbers). S1p(3; n,m)= A046089(n,m), S1p(4; n,m)= A049352, S1p(5; n,m)= A049353(n,m), S1p(6; n,m)= A049374(n, m).
The Bell transform of factorial(n+6)/factorial(6). For the definition of the Bell transform see A264428. - Peter Luschny, Jan 18 2016
Examples
{1}; {7,1}; {56,21,1}; {504,371,42,1}; ... E.g. Row polynomial E(3,x)=56*x+21*x^2+x^3.
a(4,2)= 371 = 4*(7*8)+3*(7*7) from the two types of unordered 2-forests of unary increasing trees associated with the two m=2 parts partitions (1,3) and (2^2) of n=4. The first type has 4 increasing labelings, each coming in (1)*(1*7*8)=56 colored versions, e.g., ((1c1),(2c1,3c7,4c5)) with lcp for vertex label l and color p. Here the vertex labeled 3 has depth j=1, hence 7 colors, c1..c7, can be chosen and the vertex labeled 4 with j=2 can come in 8 colors, e.g., c1..c8. Therefore there are 4*((1)*(1*7*8))=224 forests of this (1,3) type. Similarly the (2,2) type yields 3*((1*7)*(1*7))=147 such forests, e.g. ((1c1,3c4)(2c1,4c7)) or ((1c1,3c6)(2c1,4c2)), etc. - _Wolfdieter Lang_, Oct 05 2007
Links
- W. Lang, On generalizations of Stirling number triangles, J. Integer Seqs., Vol. 3 (2000), #00.2.4.
- W. Lang, First ten rows.
Programs
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Maple
# The function BellMatrix is defined in A264428. # Adds (1,0,0,0, ..) as column 0. BellMatrix(n -> (n+6)!/6!, 9); # Peter Luschny, Jan 27 2016
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Mathematica
a[n_, m_] /; n >= m >= 1 := a[n, m] = (6*m + n - 1)*a[n-1, m] + a[n-1, m-1]; a[n_, m_] /; n < m = 0; a[, 0] = 0; a[1, 1] = 1; Flatten[Table[a[n, m], {n, 1, 9}, {m, 1, n}]][[1 ;; 39]] (* _Jean-François Alcover, Jun 01 2011, after formula *) 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[(# + 6)!/6! &, rows]; Table[M[[n, k]], {n, 2, rows}, {k, 2, n}] // Flatten (* Jean-François Alcover, Jun 24 2018, after Peter Luschny *)
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Sage
# uses[bell_matrix from A264428] # Adds a column 1,0,0,0, ... at the left side of the triangle. bell_matrix(lambda n: factorial(n+6)/factorial(6), 10) # Peter Luschny, Jan 18 2016
Formula
a(n, m) = n!*A132166(n, m)/(m!*6^(n-m)); a(n, m) = (6*m+n-1)*a(n-1, m) + a(n-1, m-1), n >= m >= 1; a(n, m)=0, n
Comments