Igor Victorovich Statsenko - cp's OEIS frontend

A381082 Triangle T(n,k) read by rows, where the columns are the coefficients of the standard expansion of the function f(x) = (-log(1-x))^(k)exp(-mx)/k! for the case m=2.

1, -2, 1, 4, -3, 1, -8, 8, -3, 1, 16, -18, 11, -2, 1, -32, 44, -20, 15, 0, 1, 64, -80, 94, 5, 25, 3, 1, -128, 272, 56, 294, 105, 49, 7, 1, 256, 112, 1868, 1596, 1169, 392, 98, 12, 1, -512, 5280, 12216, 16148, 10290, 4305, 1092, 186, 18, 1

Offset: 0

Igor Victorovich Statsenko, Feb 13 2025

			Triangle starts:
  [0]     1;
  [1]    -2,      1;
  [2]     4,     -3,       1;
  [3]    -8,      8,      -3,       1;
  [4]    16,    -18,      11,      -2,       1;
  [5]   -32,     44,     -20,      15,       0,        1;
  [6]    64,    -80,      94,       5,      25,        3,     1;
  [7]  -128,    272,      56,     294,     105,       49,     7,     1;
  [8]   256,    112,    1868,    1596,    1169,      392,    98,    12,    1;
  [9]  -512,   5280,   12216,   16148,   10290,     4305,  1092,   186,   18,     1;
  ...

Cf. A000023 (row sums).
Columns 0,1: A122803, A346397.
Triangles: for m = -3 is A327997; for m = -2 is A137346 (unsigned); for m = -1 is A094816; for m = 0 is A132393; for m = 1 is A269953.

T:=(m,n,k)->add(Stirling1(n-i,k)*binomial(n,i)*m^(i)*(-1)^(n-k), i=0..n):
m:=2:seq(print(seq(T(m,n,k), k=0..n)), n=0..9);

T(n,k) = Sum_{i=0..n} Stirling1(n-i, k)*binomial(n, i)*m^(i)*(-1)^(n-k), where m = 2.

A380851 Riordan array ((1-x)^(m-1), x/(1-x)) with factor r^(2*n) on row n, for m = 3/2, r = 2.

Original entry on oeis.org

1, -2, 4, -2, 8, 16, -4, 24, 96, 64, -10, 80, 480, 640, 256, -28, 280, 2240, 4480, 3584, 1024, -84, 1008, 10080, 26880, 32256, 18432, 4096, -264, 3696, 44352, 147840, 236544, 202752, 90112, 16384, -858, 13728, 192192, 768768, 1537536, 1757184, 1171456, 425984, 65536

Offset: 0

Igor Victorovich Statsenko, Feb 06 2025

			Triangle starts:
       k = 0      1       2        3        4        5       6
  n=0:     1;
  n=1:    -2,     4;
  n=2:    -2,     8,     16;
  n=3:    -4,    24,     96,      64;
  n=4:   -10,    80,    480,     640,     256;
  n=5:   -28,   280,   2240,    4480,    3584,    1024;
  n=6:   -84,  1008,  10080,   26880,   32256,   18432,   4096;

Igor Victorovich Statsenko, Identification of Riordan generalizations of binomial coefficients, Innovation science No 02-1, State Ufa, Aeterna Publishing House, 2025, pp. 11-18, see table 10. In Russian.

Columns: A002420 (k=0); A240530 (k=1).

Triangle for m=-3, r=1: A104713; for m=-2, r=1: A104712; for m=-1, r=1: A135278; for m=0, r=1: A007318; for m=1, r=1: A097805; for m=2, r=1: A159854.

T:=(m,r,n,k)->add(binomial(i+m,m)*binomial(n+1,n-k-i)*r^(2*n)*(-1)^(i),i=0..n-k): m:=3/2: r:=2: seq(print(seq(T(m,r,n,k), k=0..n)), n=0..10);

T[n_, k_] := 4^n Binomial[n, k] Hypergeometric2F1[3/2, k - n, k + 1, 1];
Table[T[n, k], {n, 0, 8}, {k, 0, n}] // Flatten  (* Peter Luschny, Feb 07 2025 *)

# Using function riordan_array from A256893.
RA = riordan_array((1 - x)^(3/2 - 1), x/(1-x), 7)
for n in range(7): print(4^n * RA.row(n)[:n+1])  # Peter Luschny, Feb 28 2025

T(n,k) = Sum_{i=0..n-k} binomial(i+m, m)*binomial(n+1, n-k-i)*r^(2*n)*(-1)^(i), for m = 3/2 and r = 2.
From Peter Luschny, Feb 07 2025: (Start)
T(n,k) = r^(2*n)*JacobiP(n - k, 1 + k, m - 1 - n, -1).
T(n,k) = 4^n*binomial(n, k)*hypergeom([3/2, k - n], [k + 1], 1). (End)

A377058 Triangle of generalized Stirling numbers of the lower level of the hierarchy (case m=2).

Original entry on oeis.org

1, 5, 1, 32, 11, 1, 248, 113, 18, 1, 2248, 1230, 263, 26, 1, 23272, 14534, 3765, 505, 35, 1, 270400, 186992, 55654, 9115, 865, 45, 1, 3479744, 2612000, 865186, 163779, 19110, 1372, 56, 1, 49079936, 39434448, 14235388, 3013164, 408569, 36288, 2058, 68, 1

Offset: 0

Igor Victorovich Statsenko, Oct 14 2024

nonn

These numbers are a subset of the generalized Stirling numbers introduced in A370518. Therefore, we assume them to be numbers of the lower level of hierarchy with respect to A370518.

			[0]           1;
[1]           5,          1;
[2]          32,         11,         1;
[3]         248,        113,        18,        1;
[4]        2248,       1230,       263,       26,       1;
[5]       23272,      14534,      3765,      505,      35,      1;
[6]      270400,     186992,     55654,     9115,     865,     45,     1;
[7]     3479744,    2612000,    865186,   163779,   19110,   1372,    56,    1;
[8]    49079936,   39434448,  14235388,  3013164,  408569,  36288,  2058,   68,    1;

Igor Victorovich Statsenko, Relationships of "P"-generalized Stirling numbers of the first kind with other generalized Stirling numbers, Innovation science No 10-1, State Ufa, Aeterna Publishing House, 2024, pp. 19-22. In Russian.

A361649 (row sums).
Triangle for m=0: A130534.
Triangle for m=1: A376863.

T := (m,n,k) -> add(add(Stirling1(n-j,k)*binomial(n+m,i)*binomial(n,j)*binomial(j,i)*i!*m^(j-i), j=i..n), i=0..n): m:=2: seq(seq(T(m,n,k), k=0..n), n=0..10);

T(m, n, k) = Sum_{i=0..n} Sum_{j=i..n} Stirling1(n-j, k)*binomial(n+m, i)*binomial(n, j)* binomial(j, i)*i!*m^(j-i), for m = 2.

A376634 Triangle read by rows: T(n, k) = Sum_{i=0..n-k} Stirling1(i + m, m)binomial(n+m+1, n-k-i)(n + m - k)!/(i + m)!, for m = 2.

Original entry on oeis.org

1, 9, 1, 71, 12, 1, 580, 119, 15, 1, 5104, 1175, 179, 18, 1, 48860, 12154, 2070, 251, 21, 1, 509004, 133938, 24574, 3325, 335, 24, 1, 5753736, 1580508, 305956, 44524, 5000, 431, 27, 1, 70290936, 19978308, 4028156, 617624, 74524, 7155, 539, 30, 1, 924118272, 270074016, 56231712, 8969148, 1139292, 117454, 9850, 659, 33, 1, 13020978816, 3894932448, 832391136, 136954044, 18083484, 1961470, 176554, 13145, 791, 36, 1

Offset: 0

Igor Victorovich Statsenko, Sep 30 2024

The columns of the triangle T(m,n,k) represent the coefficients of the asymptotic expansion of the higher order exponential integral E(x,m+1,k+2), for m=2, k>=0. For reference see. A163931.

			Triangle starts:
 [0]          1;
 [1]          9,          1;
 [2]         71,         12,          1;
 [3]        580,        119,         15,        1;
 [4]       5104,       1175,        179,       18,        1;
 [5]      48860,      12154,       2070,      251,       21,       1;
 [6]     509004,     133938,      24574,     3325,      335,      24,     1;
 [7]    5753736,    1580508,     305956,    44524,     5000,     431,    27,     1;

Igor Victorovich Statsenko, Relationships of P-generalized Stirling numbers of the first kind with other generalized Stirling numbers, Innovation science No 10-1, State Ufa, Aeterna Publishing House, 2024, pp. 19-22. In Russian.

Column k: A001706 (k=0), A001712 (k=1), A001717 (k=2), A001722 (k=3), A051525 (k=4), A051546 (k=5), A051561 (k=6).

Cf. A094587 and A173333 for m=0, A376582 for m=1.

T:=(m,n,k)->add(Stirling1(i+m,m)*binomial(n+m+1,n-k-i)*(n+m-k)!/(i+m)!,i=0..n-k):m:=2:seq(seq(T(m,n,k), k=0..n),n=0..10);

A376863 Triangle of generalized Stirling numbers of the lower level of the hierarchy (section m=1).

Original entry on oeis.org

1, 3, 1, 13, 7, 1, 73, 50, 12, 1, 501, 400, 125, 18, 1, 4051, 3609, 1335, 255, 25, 1, 37633, 36463, 15214, 3485, 460, 33, 1, 394353, 408694, 186949, 48769, 7805, 763, 42, 1, 4596553, 5036792, 2479602, 714364, 131299, 15708, 1190, 52, 1, 58941091, 67714809, 35419350, 11045558, 2256933, 312375, 29190, 1770, 63, 1, 824073141, 986271823, 543025851, 180766890, 40194965, 6221397, 676893, 50970, 2535, 75, 1

Offset: 0

Igor Victorovich Statsenko, Oct 07 2024

			Triangle starts:
[0]        1;
[1]        3,        1;
[2]       13,        7,        1;
[3]       73,       50,       12,       1;
[4]      501,      400,      125,      18,       1;
[5]     4051,     3609,     1335,     255,      25,       1;
[6]    37633,    36463,    15214,    3485,     460,      33,      1;
[7]   394353,   408694,   186949,   48769,    7805,     763,     42,    1;
[8]  4596553,  5036792,  2479602,  714364,  131299,   15708,   1190,   52,     1;

Igor Victorovich Statsenko, Relationships of “P”-generalized Stirling numbers of the first kind with other generalized Stirling numbers, Innovation science No 10-1, State Ufa, Aeterna Publishing House, 2024, pp. 19-12. In Russian.

A000262 (column 0), A052852 (row sums).

Triangle for m=0: A130534.

T:=(m,n,k)->add(add(Stirling1(n-j,k)*binomial(n+m,i)*binomial(n,j)*binomial(j,i)*i!*m^(j-i), j=i..n),i=0..n):m:=1:seq(seq(T(m,n,k),k=0..n),n=0..10);

T(m, n, k) = Sum_{i=0..n} Sum_{j=i..n} Stirling1(n-j, k) * binomial(n+m, i) * binomial(n, j)* binomial(j, i) * i! * m^(j - i), for m = 1.

A376582 Triangle of generalized Stirling numbers.

Original entry on oeis.org

1, 5, 1, 26, 7, 1, 154, 47, 9, 1, 1044, 342, 74, 11, 1, 8028, 2754, 638, 107, 13, 1, 69264, 24552, 5944, 1066, 146, 15, 1, 663696, 241128, 60216, 11274, 1650, 191, 17, 1, 6999840, 2592720, 662640, 127860, 19524, 2414, 242, 19, 1, 80627040, 30334320, 7893840, 1557660, 245004, 31594, 3382, 299, 21, 1

Offset: 0

Igor Victorovich Statsenko, Sep 29 2024

			Triangle starts:
[0]       1;
[1]       5,       1;
[2]      26,       7,       1;
[3]     154,      47,       9,        1;
[4]    1044,     342,      74,       11,       1;
[5]    8028,    2754,     638,      107,      13,     1;
[6]   69264,   24552,    5944,     1066,     146,    15,    1;
[7]  663696,  241128,   60216,    11274,    1650,   191,   17,    1;

Column k: A001705 (k=0), A001711 (k=1), A001716 (k=2), A001721 (k=3), A051524 (k=4), A051545 (k=5), A051560 (k=6).

Cf. A094587 and A173333 for m=0.

T:=(m,n,k)->add(Stirling1(i+m,m)*binomial(n+m+1,n-k-i)*(n+m-k)!/(i+m)!,i=0..n-k): m:=1: seq(seq(T(m,n,k), k=0..n), n=0..10);

T(m,n,k) = Sum_{i=0..n-k} Stirling1(i+m,m)*binomial(n+m+1,n-k-i)*(n+m-k)!/(i+m)!, for m=1.

A370518 Triangle of numbers read by rows: T(n,k) = Sum_{i=0..n} binomial(n,i)(n-i)!Stirling1(i,k)TC(n,i) where TC(n,k) = Sum_{i=0..n-k} binomial(n+1,n-k-i)Stirling2(i+3,i+1)*(-1)^(i) for n >= 0, 0 <= k <= n.

Original entry on oeis.org

1, -5, 1, 14, -9, 1, -18, 29, -12, 1, 0, -22, 35, -14, 1, 0, -26, 15, 25, -15, 1, 0, -60, 4, 75, -5, -15, 1, 0, -204, -56, 259, 70, -56, -14, 1, 0, -912, -484, 1092, 609, -168, -126, -12, 1, 0, -5040, -3708, 5480, 4599, -231, -882, -210, -9, 1, 0, -33120, -30024, 31820, 36350, 3675, -6027, -2370, -300, -5, 1

Offset: 0

Igor Victorovich Statsenko, Feb 21 2024

Generalized Stirling numbers of the first kind of the second order.

			n\k     0     1     2     3     4     5     6
0:      1
1:     -5     1
2:     14    -9     1
3:    -18    29   -12     1
4:      0   -22    35   -14     1
5:      0   -26    15    25   -15     1
6:      0   -60     4    75    -5   -15     1

Igor Victorovich Statsenko, On the ordinal numbers of triangles of generalized special numbers, Innovation science No 2-2, State Ufa, Aeterna Publishing House, 2024, pp. 15-19. In Russian.

For m=0 the formula gives the sequence A130534; for m=1 the formula gives the sequence A094645. In this case, we assume that A130534 consists of generalized Stirling numbers of the first kind of zero order, and A094645 consists of generalized Stirling numbers of the first kind of the first order.

C:=(n,k)->n!/(k!*(n-k)!) : T0:=(m,n,k)->sum(C(n+1,n-k-p)*Stirling2(p+m+1,p+1)*((-1)^p), p=0..n-k) : T:=(m,n,k)->sum(C(n,r)*(n-r)!*Stirling1(r,k)*T0(m,n,r), r=0..n)  m:=2 : seq(seq T(m,n,k), k=0..n), n=0..10);

T(n,k) = Sum_{i=0..n} binomial(n,i)*(n-i)!*Stirling1(i,k)*TC(m,n,i) where TC(m,n,k) = Sum_{i=0..n-k} binomial(n+1,n-k-i)*Stirling2(i+m+1,i+1)*(-1)^(i),m = 2 for n >= 0.

A370516 Triangle of numbers read by rows: T(n,k) = Sum_{i=0..n-k} binomial(n+1,n-k-i)Stirling2(i+3,i+1)(-1)^i for n >= 0, 0 <= k <= n.

Original entry on oeis.org

1, -5, 1, 7, -4, 1, -3, 3, -3, 1, 0, 0, 0, -2, 1, 0, 0, 0, -2, -1, 1, 0, 0, 0, -2, -3, 0, 1, 0, 0, 0, -2, -5, -3, 1, 1, 0, 0, 0, -2, -7, -8, -2, 2, 1, 0, 0, 0, -2, -9, -15, -10, 0, 3, 1, 0, 0, 0, -2, -11, -24, -25, -10, 3, 4

Offset: 0

Igor Victorovich Statsenko, Feb 21 2024

Generalized binomial coefficients of the second order.

			n\k   0    1    2    3    4    5    6
0:    1
1:   -5    1
2:    7   -4    1
3:   -3    3   -3    1
4:    0    0    0   -2    1
5:    0    0    0   -2   -1    1
6:    0    0    0   -2   -3    0    1

Igor Victorovich Statsenko, On the ordinal numbers of triangles of generalized special numbers, Innovation science No 2-2, State Ufa, Aeterna Publishing House, 2024, pp. 15-19. In Russian.

For m=0 the formula gives the sequence A007318; for m=1 the formula gives the sequence A159854. In this case, we assume that A007318 consists of generalized binomial coefficients of order zero and A159854 consists of generalized binomial coefficients of order one.

C:=(n,k)->n!/(k!*(n-k)!) : T:=(m,n,k)->sum(C(n+1,n-k-r)*Stirling2(r+m+1,r+1)*((-1)^r), r=0..n-k) : m:=2 : seq(seq T(m,n,k), k=0..n), n=0..10);

T(n,k) = Sum_{i=0..n-k} binomial(n+1,n-k-i)*Stirling2(i+m+1,i+1)*(-1)^i where m = 2 for n >= 0, 0 <= k <= n.

A369381 Triangle of numbers read by rows T(n,k) = binomial(n+1,k+1)*Stirling2(n+k,k).

Original entry on oeis.org

1, 0, 1, 0, 3, 7, 0, 6, 60, 90, 0, 10, 310, 1505, 1701, 0, 15, 1260, 14490, 46620, 42525, 0, 21, 4445, 105875, 716205, 1727110, 1323652, 0, 28, 14280, 653100, 8162000, 38623200

Offset: 0

Igor Victorovich Statsenko, Jan 22 2024

The triangle T(n,k) is a functional dual of the triangle A269939 in identity: B(n) = Sum_{k=0..n}(-1)^(k)*A269939(n,k)/Binomial(n+k,k) = Sum_{k=0..n}(-1)^(k)*T(n,k)/Binomial(n+k,k). Where B(n) are the Bernoulli numbers.

			n\k  0      1       2        3        4       5
0:    1
1:    0      1
2:    0      3       7
3:    0      6      60       90
4:    0     10     310     1505     1701
5:    0     15    1260    14490    46620    42525

I. V. Statsenko, Functional twin of triangle A269939, Innovation science No 1-2, State Ufa, Aeterna Publishing House, 2024, pp. 15-19. In Russian.

Cf. A007820 (right diagonal).

T:=(n,k)->((n+1)!/((k+1)!*(n-k)!))*Stirling2(n+k,k):seq(seq T(n,k),k=0..n), n=0..10);

T(n,k) = binomial(n+1,k+1)*Stirling2(n+k,k).

A362791 Triangle of numbers read by rows, T(n, k) = (n(n-1)(n-2))*Stirling2(k, 3), for n >= 1 and 1 <= k <= n.

Original entry on oeis.org

0, 0, 0, 0, 0, 6, 0, 0, 24, 144, 0, 0, 60, 360, 1500, 0, 0, 120, 720, 3000, 10800, 0, 0, 210, 1260, 5250, 18900, 63210, 0, 0, 336, 2016, 8400, 30240, 101136, 324576, 0, 0, 504, 3024, 12600, 45360, 151704, 486864, 1524600, 0, 0, 720, 4320, 18000, 64800, 216720, 695520, 2178000, 6717600

Offset: 1

Igor Victorovich Statsenko, May 04 2023

T(n, k) is the number of ways to distribute k labeled items into n labeled boxes so that there are exactly 3 nonempty boxes.

			n\k   1      2      3      4      5      6      7
1:    0
2:    0      0
3:    0      0      6
4:    0      0     24    144
5:    0      0     60    360   1500
6:    0      0    120    720   3000  10800
7:    0      0    210   1260   5250  18900  63210
  ...
T(4,3) = 24:  {1}{2}{3}{} (24 ways).
T(4,4) = 144: {12}{3}{4}{} (144 ways).

I. V. Statsenko, Generalized layout problem, Innovation science No 4-2, State Ufa, Aeterna Publishing House, 2023, pp. 10-13. In Russian.

Cf. A002024 (case L=1), A362685 (case L=2), A068605 (right diagonal).

L := 3: T := (n, k) -> pochhammer(-n, L)*Stirling2(k, L)*((-1)^L):
seq(seq(T(n, k), k = 1..n), n = 1..10);

from math import isqrt, comb
from sympy.functions.combinatorial.numbers import stirling
def A362791(n): return (a:=(m:=isqrt(k:=n<<1))+(k>m*(m+1)))*(a-1)*(a-2)*stirling(n-comb(a,2),3) # Chai Wah Wu, Jun 20 2025

T(n, k) = (n!/(n - L)!) * Stirling2(k, L) with L = 3, T(1,1)=T(2,1)=T(2,2) = 0.

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User: Igor Victorovich Statsenko

A381082 Triangle T(n,k) read by rows, where the columns are the coefficients of the standard expansion of the function f(x) = (-log(1-x))^(k)*exp(-m*x)/k! for the case m=2.

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A380851 Riordan array ((1-x)^(m-1), x/(1-x)) with factor r^(2*n) on row n, for m = 3/2, r = 2.

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A377058 Triangle of generalized Stirling numbers of the lower level of the hierarchy (case m=2).

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A376634 Triangle read by rows: T(n, k) = Sum_{i=0..n-k} Stirling1(i + m, m)*binomial(n+m+1, n-k-i)*(n + m - k)!/(i + m)!, for m = 2.

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A376863 Triangle of generalized Stirling numbers of the lower level of the hierarchy (section m=1).

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A376582 Triangle of generalized Stirling numbers.

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A370518 Triangle of numbers read by rows: T(n,k) = Sum_{i=0..n} binomial(n,i)*(n-i)!*Stirling1(i,k)*TC(n,i) where TC(n,k) = Sum_{i=0..n-k} binomial(n+1,n-k-i)*Stirling2(i+3,i+1)*(-1)^(i) for n >= 0, 0 <= k <= n.

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A370516 Triangle of numbers read by rows: T(n,k) = Sum_{i=0..n-k} binomial(n+1,n-k-i)*Stirling2(i+3,i+1)*(-1)^i for n >= 0, 0 <= k <= n.

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A381082 Triangle T(n,k) read by rows, where the columns are the coefficients of the standard expansion of the function f(x) = (-log(1-x))^(k)exp(-mx)/k! for the case m=2.

A376634 Triangle read by rows: T(n, k) = Sum_{i=0..n-k} Stirling1(i + m, m)binomial(n+m+1, n-k-i)(n + m - k)!/(i + m)!, for m = 2.

A370518 Triangle of numbers read by rows: T(n,k) = Sum_{i=0..n} binomial(n,i)(n-i)!Stirling1(i,k)TC(n,i) where TC(n,k) = Sum_{i=0..n-k} binomial(n+1,n-k-i)Stirling2(i+3,i+1)*(-1)^(i) for n >= 0, 0 <= k <= n.

A370516 Triangle of numbers read by rows: T(n,k) = Sum_{i=0..n-k} binomial(n+1,n-k-i)Stirling2(i+3,i+1)(-1)^i for n >= 0, 0 <= k <= n.

A362791 Triangle of numbers read by rows, T(n, k) = (n(n-1)(n-2))*Stirling2(k, 3), for n >= 1 and 1 <= k <= n.