A049460 -id:A049460 - cp's OEIS frontend

A001715 a(n) = n!/6.

1, 4, 20, 120, 840, 6720, 60480, 604800, 6652800, 79833600, 1037836800, 14529715200, 217945728000, 3487131648000, 59281238016000, 1067062284288000, 20274183401472000, 405483668029440000, 8515157028618240000, 187333454629601280000, 4308669456480829440000

Offset: 3

N. J. A. Sloane

The numbers (4, 20, 120, 840, 6720, ...) arise from the divisor values in the general formula a(n) = n*(n+1)*(n+2)*(n+3)* ... *(n+k)*(n*(n+k) + (k-1)*k/6)/((k+3)!/6) (which covers the following sequences: A000578, A000537, A024166, A101094, A101097, A101102). - Alexander R. Povolotsky, May 17 2008
a(n) is also the number of decreasing 3-cycles in the decomposition of permutations as product of disjoint cycles, a(3)=1, a(4)=4, a(5)=20. - Wenjin Woan, Dec 21 2008
Equals eigensequence of triangle A130128 reflected. - Gary W. Adamson, Dec 23 2008
a(n) is the number of n-permutations having 1, 2, and 3 in three distinct cycles. - Geoffrey Critzer, Apr 26 2009
From Johannes W. Meijer, Oct 20 2009: (Start)
The asymptotic expansion of the higher order exponential integral E(x,m=1,n=4) ~ exp(-x)/x*(1 - 4/x + 20/x^2 - 120/x^3 + 840/x^4 - 6720/x^5 + 60480/x^6 - 604800/x^7 + ...) leads to the sequence given above. See A163931 and A130534 for more information.
(End)

N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Vincenzo Librandi, Table of n, a(n) for n = 3..200
Somaya Barati, Beáta Bényi, Abbas Jafarzadeh, and Daniel Yaqubi, Mixed restricted Stirling numbers, arXiv:1812.02955 [math.CO], 2018.
INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 263.
Wolfdieter Lang, On generalizations of Stirling number triangles, J. Integer Seqs., Vol. 3 (2000), Article 00.2.4.
D. S. Mitrinovic and R. S. Mitrinovic, Tableaux d'une classe de nombres reliés aux nombres de Stirling, Univ. Beograd. Publ. Elektrotehn. Fak. Ser. Mat. Fiz. No. 77 1962, 77 pp.
Alexsandar Petojevic, The Function vM_m(s; a; z) and Some Well-Known Sequences, Journal of Integer Sequences, Vol. 5 (2002), Article 02.1.7.
A. N. Stokes, Continued fraction solutions of the Riccati equation, Bull. Austral. Math. Soc. Vol. 25 (1982), 207-214.
Index entries for sequences related to factorial numbers.
Index to divisibility sequences.

Cf. A049352, A049458, A049460.
Cf. A034472, A130128.
Cf. A245334, A000142, A111530.
Cf. A001710, A001720, A007759, A094638.
Cf. A094587, A138533, A173333, A213936.

a001715 = (flip div 6) . a000142 -- Reinhard Zumkeller, Aug 31 2014

[Factorial(n)/6: n in [3..30]]; // Vincenzo Librandi, Jun 20 2011

f := proc(n) n!/6; end;
BB:= [S, {S = Prod(Z,Z,C), C = Union(B,Z,Z), B = Prod(Z,C)}, labelled]: seq(combstruct[count](BB, size=n)/12, n=3..20); # Zerinvary Lajos, Jun 19 2008
G(x):=1/(1-x)^4: f[0]:=G(x): for n from 1 to 18 do f[n]:=diff(f[n-1],x) od: x:=0: seq(f[n],n=0..16); # Zerinvary Lajos, Apr 01 2009

a[n_]:=n!/6; (*Vladimir Joseph Stephan Orlovsky, Dec 13 2008 *)
Range[3,30]!/6 (* Harvey P. Dale, Aug 12 2012 *)

a(n)=n!/6 \\ Charles R Greathouse IV, Jan 12 2012

a(n) = A049352(n-2, 1) (first column of triangle).
E.g.f. if offset 0: 1/(1-x)^4.
a(n) = A173333(n,3). - Reinhard Zumkeller, Feb 19 2010
G.f.: G(0)/2, where G(k) = 1 + 1/(1 - x/(x + 1/(k+4)/G(k+1))); (continued fraction). - Sergei N. Gladkovskii, Jun 01 2013
G.f.: W(0), where W(k) = 1 - x*(k+4)/( x*(k+4) - 1/(1 - x*(k+1)/( x*(k+1) - 1/W(k+1) ))); (continued fraction). - Sergei N. Gladkovskii, Aug 26 2013
a(n) = A245334(n,n-3) / 4. - Reinhard Zumkeller, Aug 31 2014
From Peter Bala, May 22 2017: (Start)
The o.g.f. A(x) satisfies the Riccati equation x^2*A'(x) + (4*x - 1)*A(x) + 1 = 0.
G.f. as an S-fraction: A(x) = 1/(1 - 4*x/(1 - x/(1 - 5*x/(1 - 2*x/(1 - 6*x/(1 - 3*x/(1 - ... - (n + 3)*x/(1 - n*x/(1 - ... ))))))))) (apply Stokes, 1982).
A(x) = 1/(1 - 3*x - x/(1 - 4*x/(1 - 2*x/(1 - 5*x/(1 - 3*x/(1 - 6*x/(1 - ... - n*x/(1 - (n+3)*x/(1 - ... ))))))))). (End)
H(x) = (1 - (1 + x)^(-3)) / 3 = x - 4 x^2/2! + 20 x^3/3! - ... is an e.g.f. of the signed sequence (n!/4!), which is the compositional inverse of G(x) = (1 - 3*x)^(-1/3) - 1, an e.g.f. for A007559. Cf. A094638, A001710 (for n!/2!), and A001720 (for n!/4!). Cf. columns of A094587, A173333, and A213936 and rows of A138533.- Tom Copeland, Dec 27 2019
E.g.f.: x^3 / (3! * (1 - x)). - Ilya Gutkovskiy, Jul 09 2021
From Amiram Eldar, Jan 15 2023: (Start)
Sum_{n>=3} 1/a(n) = 6*e - 15.
Sum_{n>=3} (-1)^(n+1)/a(n) = 3 - 6/e. (End)

More terms from Harvey P. Dale, Aug 12 2012

A001725 a(n) = n!/5!.

Original entry on oeis.org

1, 6, 42, 336, 3024, 30240, 332640, 3991680, 51891840, 726485760, 10897286400, 174356582400, 2964061900800, 53353114214400, 1013709170073600, 20274183401472000, 425757851430912000, 9366672731480064000, 215433472824041472000, 5170403347776995328000

Offset: 5

N. J. A. Sloane

The asymptotic expansion of the higher-order exponential integral E(x,m=1,n=6) ~ exp(-x)/x*(1 - 6/x + 42/x^2 - 336/x^3 + 3024/x^4 - 30240/x^5 + 332640/x^6 - 3991680/x^7 + ...) leads to the sequence given above. See A163931 and A130534 for more information. - Johannes W. Meijer, Oct 20 2009

N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Vincenzo Librandi, Table of n, a(n) for n = 5..300
INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 265.
Wolfdieter Lang, On generalizations of Stirling number triangles, J. Integer Seqs., Vol. 3 (2000), Article 00.2.4.
D. S. Mitrinovic and R. S. Mitrinovic, Tableaux d'une classe de nombres reliés aux nombres de Stirling. II, Univ. Beograd. Publ. Elektrotehn. Fak. Ser. Mat. Fiz. No. 107-108 1963 1-77.
Alexsandar Petojevic, The Function vM_m(s; a; z) and Some Well-Known Sequences, Journal of Integer Sequences, Vol. 5 (2002), Article 02.1.7
Index entries for sequences related to factorial numbers.
Index to divisibility sequences.

a(n)= A049374(n-4), n >= 1 (first column of triangle). Cf. A049460, A051339. a(n)= A051338(n-5, 0)*(-1)^(n-1) (first unsigned column of triangle).

Cf. A245334, A000142.

a001725 = (flip div 120) . a000142 -- Reinhard Zumkeller, Aug 31 2014

[Factorial(n)/120: n in [5..25]]; // Vincenzo Librandi, Jul 20 2011

lst={};Do[AppendTo[lst, n!/5! ], {n, 5, 5!}];lst (* Vladimir Joseph Stephan Orlovsky, Oct 25 2008 *)
Range[5,30]!/120 (* Harvey P. Dale, Dec 20 2014 *)

a(n)=n!/120 \\ Charles R Greathouse IV, Jul 19 2011

E.g.f. if offset 0: 1/(1-x)^6.
a(n) = A173333(n,5). - Reinhard Zumkeller, Feb 19 2010
G.f.: G(0)/2, where G(k)= 1 + 1/(1 - x*(k+6)/(x*(k+6) + 1/G(k+1))); (continued fraction). - Sergei N. Gladkovskii, Jun 06 2013
G.f.: W(0)/(40*x^2) -1/(20*x^2) -1/(5*x) , where W(k) = 1 + 1/( 1 - x*(k+4)/( x*(k+4) + 1/W(k+1) )); (continued fraction). - Sergei N. Gladkovskii, Aug 21 2013
a(n) = A245334(n,n-5) / 6. - Reinhard Zumkeller, Aug 31 2014
E.g.f.: x^5 / (5! * (1 - x)). - Ilya Gutkovskiy, Jul 09 2021
From Amiram Eldar, Jan 15 2023: (Start)
Sum_{n>=5} 1/a(n) = 120*e - 325.
Sum_{n>=5} (-1)^(n+1)/a(n) = 45 - 120/e. (End)

More terms from Harvey P. Dale, Dec 20 2014

A001552 a(n) = 1^n + 2^n + ... + 5^n.

Original entry on oeis.org

5, 15, 55, 225, 979, 4425, 20515, 96825, 462979, 2235465, 10874275, 53201625, 261453379, 1289414505, 6376750435, 31605701625, 156925970179, 780248593545, 3883804424995, 19349527020825, 96470431101379, 481245667164585, 2401809362313955, 11991391850823225

Offset: 0

N. J. A. Sloane

a(n)*(-1)^n, n>=0, gives the z-sequence for the Sheffer triangle A049460 ((signed) 5-restricted Stirling1 numbers), which is the inverse triangle of A193685 (5-restricted Stirling2 numbers). See the W. Lang link under A006232 for a- and z-sequences for Sheffer matrices. The a-sequence for each (signed) r-restricted Stirling1 Sheffer triangle is A027641/A027642 (Bernoulli numbers). - Wolfdieter Lang, Oct 10 2011

M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards Applied Math. Series 55, 1964 (and various reprintings), p. 813.
N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

T. D. Noe, Table of n, a(n) for n = 0..200
M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards, Applied Math. Series 55, Tenth Printing, 1972 [alternative scanned copy].
INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 365
Simon Plouffe, Approximations de séries génératrices et quelques conjectures, Dissertation, Université du Québec à Montréal, 1992; arXiv:0911.4975 [math.NT], 2009.
Simon Plouffe, 1031 Generating Functions, Appendix to Thesis, Montreal, 1992
Index entries for linear recurrences with constant coefficients, signature (15, -85, 225, -274, 120).

Column 5 of array A103438.

Table[Total[Range[5]^n], {n, 0, 40}] (* T. D. Noe, Oct 10 2011 *)

PARI
```
a(n)=if(n<0,0,sum(k=1,5,k^n))
```

[3**n + sigma(4, n) + 5**n for n in range(22)] # Zerinvary Lajos, Jun 04 2009

[1 + 2**n + 3**n + 4**n + 5**n for n in range(22)] # Zerinvary Lajos, Jun 04 2009

a(n) = Sum_{k=1..5} k^n, n >= 0.
O.g.f.: (5 - 60*x + 255*x^2 - 450*x^3 + 274*x^4)/Product_{j=1..5} (1 - j*x). - Simon Plouffe in his 1992 dissertation
E.g.f.: exp(x)*(1-exp(5*x))/(1-exp(x)) = Sum_{j=1..5} exp(j*x) (trivial). - Wolfdieter Lang, Oct 10 2011

A049459 Generalized Stirling number triangle of first kind.

Original entry on oeis.org

1, -4, 1, 20, -9, 1, -120, 74, -15, 1, 840, -638, 179, -22, 1, -6720, 5944, -2070, 355, -30, 1, 60480, -60216, 24574, -5265, 625, -39, 1, -604800, 662640, -305956, 77224, -11515, 1015, -49, 1, 6652800, -7893840, 4028156, -1155420, 203889

Offset: 0

Wolfdieter Lang

a(n,m)= ^4P_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-(4+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(4*t),exp(t)-1).
See A143493 for the unsigned version of this array and A143496 for the inverse. - Peter Bala, Aug 25 2008

			   1;
  -4,    1;
  20,   -9,   1;
-120,   74, -15,   1;
840, -638, 179, -22, 1;

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.

Reinhard Zumkeller, Rows n = 0..125 of triangle, flattened

Unsigned column sequences are: A001715-A001719. Cf. A008275 (Stirling1 triangle), A049458, A049460. Row sums (signed triangle): A001710(n+2)*(-1)^n. Row sums (unsigned triangle): A001720(n+4).
Cf. A000035 A084938.
A143493, A143496. - Peter Bala, Aug 25 2008

a049459 n k = a049459_tabl !! n !! k
a049459_row n = a049459_tabl !! n
a049459_tabl = map fst $ iterate (\(row, i) ->
   (zipWith (-) ([0] ++ row) $ map (* i) (row ++ [0]), i + 1)) ([1], 4)
-- Reinhard Zumkeller, Mar 11 2014

A049459_row := n -> seq((-1)^(n-k)*coeff(expand(pochhammer(x+4, n)), x, k), k=0..n): seq(print(A049459_row(n)),n=0..8); # Peter Luschny, May 16 2013

a[n_, m_] /; 0 <= m <= n := a[n, m] = a[n-1, m-1] - (n+3)*a[n-1, m];
a[n_, m_] /; n < m = 0;
a[_, -1] = 0; a[0, 0] = 1;
Table[a[n, m], {n, 0, 10}, {m, 0, n}] // Flatten (* Jean-François Alcover, Jun 19 2018 *)

a(n, m)= a(n-1, m-1) - (n+3)*a(n-1, m), n >= m >= 0; a(n, m) := 0, n

Triangle (signed) = [ -4, -1, -5, -2, -6, -3, -7, -4, -8, ...] DELTA [1, 0, 1, 0, 1, 0, 1, 0, ...]; triangle (unsigned) = [4, 1, 5, 2, 6, 3, 7, 4, 8, 5, ...] DELTA [1, 0, 1, 0, 1, 0, 1, 0, 1, 0, ...]; where DELTA is Deléham's operator defined in A084938 (unsigned version in A143493).

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,4), for n=1,2,...;i=0...n. - Milan Janjic, Dec 21 2008

Second formula corrected by Philippe Deléham, Nov 09 2008

A051523 Generalized Stirling number triangle of first kind.

Original entry on oeis.org

1, -10, 1, 110, -21, 1, -1320, 362, -33, 1, 17160, -6026, 791, -46, 1, -240240, 101524, -17100, 1435, -60, 1, 3603600, -1763100, 358024, -38625, 2335, -75, 1, -57657600, 31813200, -7491484, 976024, -75985, 3535, -91, 1, 980179200, -598482000, 159168428, -24083892, 2267769, -136080, 5082, -108, 1

Offset: 0

Wolfdieter Lang

a(n,m)= ^10P_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-(10+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(10*t),exp(t)-1).

			{1}; {-10,1}; {110,-21,1}; {-1320,362,-331}; ... s(2,x)= 110-21*x+x^2; S1(2,x)= -x+x^2 (Stirling1).

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).

The first (m=0) unsigned column sequence is A049398. Row sums (signed triangle): A049389(n)*(-1)^n. Row sums (unsigned triangle): A051431(n).
See also A049444 A049458 A049459 A049460 A051338 A051339 A051379 A051380
Cf. A000035 A084938.

a051523 n k = a051523_tabl !! n !! k
a051523_row n = a051523_tabl !! n
a051523_tabl = map fst $ iterate (\(row, i) ->
   (zipWith (-) ([0] ++ row) $ map (* i) (row ++ [0]), i + 1)) ([1], 10)
-- Reinhard Zumkeller, Mar 12 2014

a[n_, m_] := Pochhammer[m + 1, n - m] SeriesCoefficient[Log[1 + x]^m/(1 + x)^10, {x, 0, n}];
Table[a[n, m], {n, 0, 8}, {m, 0, n}] // Flatten (* Jean-François Alcover, Oct 29 2019 *)

a(n, m)= a(n-1, m-1) - (n+9)*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)^10).

Triangle (signed) = [ -10, -1, -11, -2, -12, -3, -13, -14, -4, ...] DELTA A000035; triangle (unsigned) = [10, 1, 11, 2, 12, 3, 13, 4, 14, 5, 15, ...] 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,10), for n=1,2,...;i=0...n. - Milan Janjic, Dec 21 2008

A087755 Triangle read by rows: Stirling numbers of the first kind (A008275) mod 2.

Original entry on oeis.org

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

Offset: 1

Philippe Deléham, Oct 02 2003

Essentially also parity of Mitrinovic's triangles A049458, A049460, A051339, A051380.

			Triangle begins:
1
1 1
0 1 1
0 1 0 1
0 0 1 0 1
0 0 1 1 1 1
0 0 0 1 1 1 1
0 0 0 1 0 0 0 1
0 0 0 0 1 0 0 0 1
0 0 0 0 1 1 0 0 1 1
0 0 0 0 0 1 1 0 0 1 1
0 0 0 0 0 1 0 1 0 1 0 1
0 0 0 0 0 0 1 0 1 0 1 0 1
0 0 0 0 0 0 1 1 1 1 1 1 1 1

Das, Sajal K., Joydeep Ghosh, and Narsingh Deo. "Stirling networks: a versatile combinatorial topology for multiprocessor systems." Discrete applied mathematics 37 (1992): 119-146. See p. 122. - N. J. A. Sloane, Nov 20 2014

p = 2; s=14; S1T = matrix(s,s,n,k, if(k==1,(-1)^(n-1)*(n-1)!)); for(n=2,s,for(k=2,n, S1T[n,k]=-(n-1)*S1T[n-1,k]+S1T[n-1,k-1]));
S1TMP = matrix(s,s,n,k, S1T[n,k]%p);
for(n=1,s,for(k=1,n,print1(S1TMP[n,k]," "));print()) /* Gerald McGarvey, Oct 17 2009 */

T(n, k) = A087748(n, k) = A008275(n, k) mod 2 = A047999([n/2], k-[(n+1)/ 2]) = T(n-2, k-2) XOR T(n-2, k-1) with T(1, 1) = T(2, 1) = T(2, 2) = 1; T(2n, k) = T(2n-1, k-1) XOR T(2n-1, k); T(2n+1, k) = T(2n, k-1). - Henry Bottomley, Dec 01 2003

Edited and extended by Henry Bottomley, Dec 01 2003

A087756 a(n) = A087745(n+1).

Original entry on oeis.org

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

Essentially a duplicate of A087745.
Mitrinovic's triangles A049458, A049460, A051339, A051380 and triangle of Stirling numbers of first kind (A008275) mod 2 converted to decimal.
See also A001317 = [1, 3, 5, 15, 17, ...].

Cf. A001317, A008275, A049458, A049460, A051339, A051380.

Edited by Omar E. Pol and N. J. A. Sloane, Dec 26 2008

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A001715 a(n) = n!/6.

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A001725 a(n) = n!/5!.

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A001552 a(n) = 1^n + 2^n + ... + 5^n.

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A049459 Generalized Stirling number triangle of first kind.

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A051523 Generalized Stirling number triangle of first kind.

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A087755 Triangle read by rows: Stirling numbers of the first kind (A008275) mod 2.

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