A008275 -id:A008275 - cp's OEIS frontend

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

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

A055924 Exponential transform of Stirling1 triangle A008275.

Original entry on oeis.org

1, -1, 2, 2, -6, 5, -6, 22, -30, 15, 24, -100, 175, -150, 52, -120, 548, -1125, 1275, -780, 203, 720, -3528, 8120, -11025, 9100, -4263, 877, -5040, 26136, -65660, 101535, -101920, 65366, -24556, 4140, 40320, -219168, 590620, -1009260, 1167348, -920808, 478842, -149040, 21147

Offset: 1

Wouter Meeussen, Christian G. Bower, Jul 06 2000

|a(n,k)| = number of sets of permutations of {1,...,n} with k total cycles.
From David Callan, Sep 20 2007: (Start)
|a(n,k)| = Stirling1(n, k) * Bell(k) counts the above sets of permutations. To see this, recall that Stirling1(n, k) is the number of permutations of [n]={1,...,n} with k cycles and Bell(k) is the number of set partitions of [k].
Given such a permutation and set partition, write the permutation in standard cycle form (smallest entry first in each cycle and first entries decreasing left to right). For example, with n=15 and k=6, {{10}, {6, 11}, {5, 7, 15}, {3, 13, 12, 8}, {2, 14, 9}, {1, 4}} is in this standard cycle form.
Then combine cycles as specified by the partition to form a set of lists. For example, the partition 156-24-3 would yield {{10, 2, 14, 9, 1, 4}, {6, 11, 3, 13, 12, 8}, {5, 7, 15}}. The original first entries are now the record left-to-right lows.
Finally, apply to each list the well known transformation that sends # record lows to # cycles. The example yields {{4, 14, 1, 2, 10, 9}, {13, 11, 3, 6, 8, 12}, {7, 15, 5}}. This is a bijection to sets of lists (i.e. permutations) with a total of k cycles, as required. (End)

			Triangle begins:
   1;
  -1,    2;
   2,   -6,   5;
  -6,   22, -30,   15;
  24, -100, 175, -150, 52;
  ...
|a(3,2)| = 6 because (12)(3), (12)|(3), (13)(2), (13)|(2), (23)(1), (23)|(1).

Row sums of |a(n,k)| give A000262.

Cf. A000110, A008275, A008297, A055925, A104150.

E.g.f.: exp((1+x)^y-1).

a(n,k) = Stirling1(n,k) * Bell(k). - Vladeta Jovovic, Feb 01 2003

A079638 Matrix product of unsigned Lah-triangle |A008297(n,k)| and unsigned Stirling1-triangle |A008275(n,k)|.

Original entry on oeis.org

1, 3, 1, 14, 9, 1, 90, 83, 18, 1, 744, 870, 275, 30, 1, 7560, 10474, 4275, 685, 45, 1, 91440, 143892, 70924, 14805, 1435, 63, 1, 1285200, 2233356, 1274196, 324289, 41160, 2674, 84, 1, 20603520, 38769840, 24870572, 7398972, 1151409, 98280, 4578

Offset: 1

Vladeta Jovovic, Jan 30 2003

Matrix product of unsigned Lah-triangle |A008297(n,k)| and Stirling1-triangle A008275(n,k) is unsigned Stirling1-triangle |A008275(n,k)|.
Also the Bell transform of A029767(n+1). For the definition of the Bell transform see A264428. - Peter Luschny, Jan 26 2016
Essentially the same as A131222. - Peter Bala, Feb 12 2022

			Triangle begins
     1;
     3,     1;
    14,     9,    1;
    90,    83,   18,   1;
   744,   870,  275,  30,  1;
  7560, 10474, 4275, 685, 45, 1;
  ...

Michael De Vlieger, Table of n, a(n) for n = 1..1225 (rows n = 1..50, flattened).
William Keith, Rishi Nath, and James Sellers, On simultaneous (s, s+t, s+2t, ...)-core partitions, arXiv:2508.00074 [math.CO], 2025. See p. 3.

Cf. A002866 (row sums), A029767 (first column), A131222.

# The function BellMatrix is defined in A264428.
# Adds (1, 0, 0, 0, ..) as column 0.
BellMatrix(n -> n!*(2^(n+1)-1), 9); # Peter Luschny, Jan 26 2016

BellMatrix[f_, len_] := With[{t = Array[f, len, 0]}, Table[BellY[n, k, t], {n, 0, len - 1}, {k, 0, len - 1}]];
B = BellMatrix[Function[n, n! (2^(n + 1) - 1)], rows = 12];
Table[B[[n, k]], {n, 2, rows}, {k, 2, n}] // Flatten (* Jean-François Alcover, Jun 28 2018, after Peter Luschny *)

T(n, k) = Sum_{i=k..n} |A008297(n, i)| * |A008275(i, k)|.

E.g.f.: ((1-x)/(1-2*x))^y. - Vladeta Jovovic, Nov 22 2003

A079639 Matrix product of Stirling1-triangle A008275(n,k) and unsigned Lah-triangle |A008297(n,k)|.

Original entry on oeis.org

1, 1, 1, 2, 3, 1, 4, 11, 6, 1, 14, 40, 35, 10, 1, 38, 184, 195, 85, 15, 1, 216, 840, 1204, 665, 175, 21, 1, 600, 4920, 7616, 5369, 1820, 322, 28, 1, 6240, 26616, 54116, 44016, 18669, 4284, 546, 36, 1, 9552, 197856, 392460, 383480, 191205, 54453, 9030, 870, 45, 1, 319296, 1177176, 3229776, 3449600, 2017070, 679371, 139293, 17490, 1320, 55, 1, -519312

Offset: 1

Vladeta Jovovic, Jan 30 2003

Also the Bell transform of A006252(n+1). For the definition of the Bell transform see A264428. - Peter Luschny, Jan 26 2016

Cf. A006252 (first column).

# The function BellMatrix is defined in A264428.
# Adds (1, 0, 0, 0, ..) as column 0.
BellMatrix(n -> add(k!*combinat:-stirling1(n+1,k),k=0..n+1),9); # Peter Luschny, Jan 26 2016

BellMatrix[f_, len_] := With[{t = Array[f, len, 0]}, Table[BellY[n, k, t], {n, 0, len - 1}, {k, 0, len - 1}]];
rows = 12;
B = BellMatrix[Function[n, Sum[k!*StirlingS1[n+1, k], {k, 0, n+1}]], rows];
Table[B[[n, k]], {n, 2, rows}, {k, 2, n}] // Flatten (* Jean-François Alcover, Jun 27 2018, after Peter Luschny *)

T(n, k) = Sum_{i=k..n} A008275(n, i) * |A008297(i, k)|.

E.g.f: (1+x)^(y/(1-log(1+x))). - Vladeta Jovovic, Nov 22 2003

A112488 Third column of triangle A112486 used for e.g.f.s of |Stirling1| = |A008275| diagonals.

Original entry on oeis.org

3, 35, 340, 3304, 33740, 367884, 4302216, 53961336, 724534272, 10386470016, 158507316864, 2567670088320, 44027031755520, 796963357981440, 15192135816261120, 304269507433658880, 6388907821376256000

Offset: 2

Wolfdieter Lang, Sep 12 2005

			340 = a(4) = 6*35 + 5*26.

G. C. Greubel, Table of n, a(n) for n = 2..445

A112486[n_, k_] :=  A112486[n, k] = Which[n < 0 || k < 0 || k > n, 0, n == 0, 1, True, (n + k)*A112486[n - 1, k] + (n + k - 1)*A112486[n - 1, k - 1]]; Table[A112486[n, 2], {n, 2, 50}] (* G. C. Greubel, Jan 21 2017 *)

a(n) = A112486(n, 2), n>=2. a(0)=0=a(1).

a(n) = (n+2)*a(n-1) + (n+1)*A001705(n-1), n>=2, a(1):=0.

A125553 Triangle read by rows: T(n,k) = S1(n,k)*2^k, where S1(n,k) is an unsigned Stirling number of the first kind (cf. A008275) (n >= 1, 1 <= k <= n).

Original entry on oeis.org

2, 2, 4, 4, 12, 8, 12, 44, 48, 16, 48, 200, 280, 160, 32, 240, 1096, 1800, 1360, 480, 64, 1440, 7056, 12992, 11760, 5600, 1344, 128, 10080, 52272, 105056, 108304, 62720, 20608, 3584, 256, 80640, 438336, 944992, 1076544, 718368, 290304, 69888, 9216, 512

Offset: 1

N. J. A. Sloane, Jan 04 2007

Row sums are factorial numbers.
T(n,k) is the number of cycle-colored n-permutations possessing exactly k cycles; two colors are available. - Steven Finch, Nov 19 2021
Subtriangle (for 1<=k<=n) of triangle given by [0,1,1,2,2,3,3,4,4,5,5,...] DELTA [2,0,2,0,2,0,2,0,2,0,...] where DELTA is the operator defined in A084938. - Philippe Deléham, Jan 05 2007
Also the Bell transform of the sequence "a(n) = 2*(n+1)!/(n+1)". For the definition of the Bell transform see A264428. - Peter Luschny, Jan 27 2016

			Triangle begins:
  2
  2 4
  4 12 8
  12 44 48 16
  48 200 280 160 32
Triangle [0,1,1,2,2,3,3,...] DELTA [2,0,2,0,2,0,2,...], for 0<=k<=n, begins:
  1;
  0, 2;
  0, 2, 4;
  0, 4, 12, 8;
  0, 12, 44, 48, 16;
  0, 48, 200, 280, 160, 32;

Steven Finch, Rounds, Color, Parity, Squares, arXiv:2111.14487 [math.CO], 2021.

Cf. A008275, A208529.

with(combinat): T:=(n,k)->2^k*abs(stirling1(n,k)): for n from 1 to 10 do seq(T(n,k),k=1..n) od; # yields sequence in triangular form - Emeric Deutsch, Jan 05 2007
A008275 := proc(n,k) if n = 0 and k = 0 then 1 ; elif n = 0 or k = 0 then 0 ; else A008275(n-1,k-1)-(n-1)*A008275(n-1,k) ; fi ; end ; A125553 := proc(n,k) abs(A008275(n,k)*2^k) ; end ; nmax := 10 ; for n from 1 to nmax do for k from 1 to n do printf("%d, ",A125553(n,k)) ; od ; od ; # R. J. Mathar, Jan 12 2007
# The function BellMatrix is defined in A264428.
# Adds (1,0,0,0, ..) as column 0.
BellMatrix(n -> 2*(n+1)!/(n+1), 9); # Peter Luschny, Jan 27 2016

Flatten[Table[Table[2^k Abs[StirlingS1[n,k]], {k,1,n}], {n,1,8}]] (* Geoffrey Critzer, Dec 14 2011 *)
BellMatrix[f_, len_] := With[{t = Array[f, len, 0]}, Table[BellY[n, k, t], {n, 0, len - 1}, {k, 0, len - 1}]];
B = BellMatrix[Function[n, 2 (n + 1)!/(n + 1)], rows = 12];
Table[B[[n, k]], {n, 2, rows}, {k, 2, n}] // Flatten (* Jean-François Alcover, Jun 28 2018, after Peter Luschny *)

E.g.f.: 1/(1-x)^(2y). - Geoffrey Critzer, Dec 14 2011

More terms from R. J. Mathar, Jan 12 2007

A104416 Triangle, read by rows, where T(n,k) = A008275(k+1,n-k+1) are Stirling numbers of the first kind.

Original entry on oeis.org

1, -1, 1, 0, -3, 1, 0, 2, -6, 1, 0, 0, 11, -10, 1, 0, 0, -6, 35, -15, 1, 0, 0, 0, -50, 85, -21, 1, 0, 0, 0, 24, -225, 175, -28, 1, 0, 0, 0, 0, 274, -735, 322, -36, 1, 0, 0, 0, 0, -120, 1624, -1960, 546, -45, 1, 0, 0, 0, 0, 0, -1764, 6769, -4536, 870, -55, 1, 0, 0, 0, 0, 0, 720, -13132, 22449, -9450, 1320, -66, 1

Offset: 0

Paul D. Hanna, Mar 06 2005

The matrix inverse forms A104417, in which column 0 equals A082161.

			A(x,y) = (1-x) + x*y*(1-x)*(1-2*x) + x^2*y^2*(1-x)*(1-2*x)*(1-3*x) + x^3*y^3*(1-x)*(1-2*x)*(1-3*x)*(1-4*x) + ...
Rows begin:
  1;
  -1,1;
  0,-3,1;
  0,2,-6,1;
  0,0,11,-10,1;
  0,0,-6,35,-15,1;
  0,0,0,-50,85,-21,1;
  0,0,0,24,-225,175,-28,1;
  0,0,0,0,274,-735,322,-36,1;
  0,0,0,0,-120,1624,-1960,546,-45,1;
  ...

Cf. A104417, A082161, A008275.

{T(n,k)=local(X=x+x*O(x^n),Y=y+y*O(y^k)); polcoeff(polcoeff(sum(i=0,n,X^i*Y^i*prod(j=1,i+1,1-j*X)),n,x),k,y)}

G.f.: A(x, y) = Sum_{n>=0} x^n*y^n*Product_{k=1..n+1} (1-k*x).

A104417 Triangle, read by rows, equal to the matrix inverse of A104416, where A104416(n,k) = A008275(k+1,n-k+1) (Stirling numbers of the first kind).

Original entry on oeis.org

1, 1, 1, 3, 3, 1, 16, 16, 6, 1, 127, 127, 49, 10, 1, 1363, 1363, 531, 115, 15, 1, 18628, 18628, 7286, 1615, 230, 21, 1, 311250, 311250, 121964, 27321, 4040, 413, 28, 1, 6173791, 6173791, 2421471, 545311, 82131, 8841, 686, 36, 1, 142190703, 142190703

Offset: 0

Paul D. Hanna, Mar 06 2005

Column 0 and column 1 contain A082161.

			Column 0 forms A082161 that satisfies:
1 = (1-x) + 1*x*(1-x)(1-2x) + 3*x^2*(1-x)(1-2x)(1-3x) +
+ 16*x^3*(1-x)(1-2x)(1-3x)(1-4x) + ...
+ A082161(n+1)*x^n*(1-x)(1-2x)(1-3x)*..*(1-(n+1)*x) + ...
this g.f. can be derived from the matrix inverse, A104416.
Rows begin:
1;
1,1;
3,3,1;
16,16,6,1;
127,127,49,10,1;
1363,1363,531,115,15,1;
18628,18628,7286,1615,230,21,1;
311250,311250,121964,27321,4040,413,28,1; ...

Cf. A104416, A082161.

PARI

A112489 Fourth column of triangle A112486 used for e.g.f.s of |Stirling1|=|A008275| diagonals.

Original entry on oeis.org

15, 315, 4900, 70532, 1008980, 14777620, 224655816, 3568061640, 59371808496, 1035987707664, 18953413075584, 363290743698048, 7287692926408704, 152811506045431296, 3344880701417587200, 76327884878442508800

Offset: 3

Wolfdieter Lang, Sep 12 2005

			4900 = a(5) = 8*315 + 7*340.

a(n)= A112486(n, 3), n>=3. a(0)=a(1)=a(2)=0.

a(n)= (n+3)*a(n-1) + (n+2)*A112488(n-1), n>=3, a(2):=0.

A112490 Fifth column of triangle A112486 used for e.g.f.s of |Stirling1|=|A008275| diagonals.

Original entry on oeis.org

105, 3465, 78750, 1571570, 29957620, 566780500, 10855452608, 212784652080, 4295131560720, 89593039854864, 1934882739672480, 43302005440341984, 1004506274408605056, 24150861883489332096, 601561456166534637312

Offset: 4

Wolfdieter Lang, Sep 12 2005

			78750 = a(6) = 10*3465 + 9*4900.

a(n)= A112486(n, 4), n>=4. a(0), ..., a(3) = 0.

a(n)= (n+4)*a(n-1) + (n+3)* A112489(n-1), n>=4, a(3):=0.

cp's OEIS Frontend

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

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A055924 Exponential transform of Stirling1 triangle A008275.

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A079638 Matrix product of unsigned Lah-triangle |A008297(n,k)| and unsigned Stirling1-triangle |A008275(n,k)|.

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A079639 Matrix product of Stirling1-triangle A008275(n,k) and unsigned Lah-triangle |A008297(n,k)|.

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A112488 Third column of triangle A112486 used for e.g.f.s of |Stirling1| = |A008275| diagonals.

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A125553 Triangle read by rows: T(n,k) = S1(n,k)*2^k, where S1(n,k) is an unsigned Stirling number of the first kind (cf. A008275) (n >= 1, 1 <= k <= n).

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A104416 Triangle, read by rows, where T(n,k) = A008275(k+1,n-k+1) are Stirling numbers of the first kind.

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A104417 Triangle, read by rows, equal to the matrix inverse of A104416, where A104416(n,k) = A008275(k+1,n-k+1) (Stirling numbers of the first kind).

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