A094587 -id:A094587 - cp's OEIS frontend

A070895 Triangle read by rows where T(n+1,k)=T(n,k)+n*T(n-1,k) starting with T(n,n)=1 and T(n,k)=0 if n

1, 1, 1, 2, 1, 1, 4, 3, 1, 1, 10, 6, 4, 1, 1, 26, 18, 8, 5, 1, 1, 76, 48, 28, 10, 6, 1, 1, 232, 156, 76, 40, 12, 7, 1, 1, 764, 492, 272, 110, 54, 14, 8, 1, 1, 2620, 1740, 880, 430, 150, 70, 16, 9, 1, 1, 9496, 6168, 3328, 1420, 636, 196, 88, 18, 10, 1, 1, 35696, 23568

Offset: 0

Henry Bottomley, May 23 2002

For n>k+1, T(n,k) is a multiple of k+2.

Eigentriangle of inverse of (-1)^(n-k)*A094587. Row sums are A187044. - Paul Barry, Mar 02 2011

			Rows start: 1; 1,1; 2,1,1; 4,3,1,1; 10,6,4,1,1; etc.
Triangle begins
1,
1, 1,
2, 1, 1,
4, 3, 1, 1,
10, 6, 4, 1, 1,
26, 18, 8, 5, 1, 1,
76, 48, 28, 10, 6, 1, 1,
232, 156, 76, 40, 12, 7, 1, 1
Production matrix begins
1, 1,
1, 0, 1,
1, 1, 0, 1,
2, 1, 1, 0, 1,
4, 3, 1, 1, 0, 1,
10, 6, 4, 1, 1, 0, 1,
26, 18, 8, 5, 1, 1, 0, 1,
76, 48, 28, 10, 6, 1, 1, 0, 1,
232, 156, 76, 40, 12, 7, 1, 1, 0, 1
Inverse begins
1,
-1, 1,
-1, -1, 1,
0, -2, -1, 1,
0, 0, -3, -1, 1,
0, 0, 0, -4, -1, 1,
0, 0, 0, 0, -5, -1, 1,
0, 0, 0, 0, 0, -6, -1, 1
- _Paul Barry_, Mar 02 2011

Columns include A000085, A000932, A059480. Right hand columns effectively include A000012 (twice), A000027, A005843, A028552. Cf. A062323 for a triangle with similar formulas.

T(n, k+1)=(T(n, k-1)-T(n-1, k))/k for 0

A348482 Triangle read by rows: T(n,k) = (Sum_{i=k..n} i!)/(k!) for 0 <= k <= n.

Original entry on oeis.org

1, 2, 1, 4, 3, 1, 10, 9, 4, 1, 34, 33, 16, 5, 1, 154, 153, 76, 25, 6, 1, 874, 873, 436, 145, 36, 7, 1, 5914, 5913, 2956, 985, 246, 49, 8, 1, 46234, 46233, 23116, 7705, 1926, 385, 64, 9, 1, 409114, 409113, 204556, 68185, 17046, 3409, 568, 81, 10, 1

Offset: 0

Werner Schulte, Oct 20 2021

The matrix inverse M = T^(-1) has terms M(n,n) = 1 for n >= 0, M(n,n-1) = -(n+1) for n > 0, and M(n,n-2) = n for n > 1, otherwise 0.

			The triangle T(n,k) for 0 <= k <= n starts:
n\k :       0       1       2      3      4     5    6   7   8  9
=================================================================
  0 :       1
  1 :       2       1
  2 :       4       3       1
  3 :      10       9       4      1
  4 :      34      33      16      5      1
  5 :     154     153      76     25      6     1
  6 :     874     873     436    145     36     7    1
  7 :    5914    5913    2956    985    246    49    8   1
  8 :   46234   46233   23116   7705   1926   385   64   9   1
  9 :  409114  409113  204556  68185  17046  3409  568  81  10  1
  etc.

Sela Fried, On a sum involving factorials, 2024.

Cf. A109398, A094587, A002104 (row sums), A173184 (alt. row sums), A000012 (main diagonal), A000027(1st subdiagonal), A000290 (2nd subdiagonal), A081437 (3rd subdiagonal), A192398 (4th subdiagonal), A003422 (column 0), A007489 (column 1), A345889 (column 2), A143122.

T[n_, k_] := Sum[i!, {i, k, n}]/k!; Table[T[n, k], {n, 0, 9}, {k, 0, n}] // Flatten (* Amiram Eldar, Oct 20 2021 *)

T(n,n) = 1 and T(2*n,n) = A109398(n) for n >= 0; T(n,n-1) = n+1 for n > 0; T(n,n-2) = n^2 for n > 1.
T(n,k) - T(n-1,k) = (n!) / (k!) = A094587(n,k) for 0 <= k < n.
T(n,k) = (k+2) * (T(n,k+1) - T(n,k+2)) for 0 <= k < n-1.
T(n,k) = (T(n,k-1) - 1) / k for 0 < k <= n.
T(n,k) * T(n-1,k-1) - T(n-1,k) * T(n,k-1) = (n!) / (k!) for 0 < k < n.
T(n,1) = T(n,0)-1 = Sum_{k=0..n-1} T(n,k)/(k+2) for n > 0 (conjectured).
Sum_{k=0..n} binomial(k+r,k) * (1-k) * T(n+r,k+r) = binomial(n+r+1,n) for n >= 0 and r >= 0.
Sum_{k=0..n} (-1)^k * (k+1) * T(n,k) = (1 + (-1)^n) / 2 for n >= 0.
Sum_{k=0..n} (-1)^k * (k!) * T(n,k) = Sum_{k=0..n} (k!) * (1+(-1)^k) / 2 for n >= 0.
The row polynomials p(n,x) = Sum_{k=0..n} T(n,k) * x^k for n >= 0 satisfy the following equations:
  (a) p(n,x) - p'(n,x) = (x^(n+1)-1) / (x-1) for n >= 0, where p' is the first derivative of p;
  (b) p(n,x) - (n+1) * p(n-1,x) + n * p(n-2,x) = x^n for n > 1.
  (c) p(n,x) = (x+1) * p(n-1,x) + 1 + Sum_{i=1..n-1} (d/dx)^i p(n-1,x) for n > 0 (conjectured).
Row sums p(n,1) equal A002104(n+1) for n >= 0.
Alternating row sums p(n,-1) equal A173184(n) for n >= 0 (conjectured).
The three conjectures stated above are true. See links. - Sela Fried, Jul 11 2024.
From Peter Luschny, Jul 11 2024: (Start)
T(n, k) = (t(k) - t(n + 1)) / k!, where t(n) = (-1)^(n + 1) * Gamma(n + 1) * Subfactorial(-(n + 1)).
T(n, k) = A143122(n, k) / k!. (End)

A352650 Triangle read by rows: T(n,k) = n * T(n-1,k) + (-1)^(n-k) for 0 <= k <= n with initial values T(n,k) = 0 if n < 0 or k < 0 or k > n.

Original entry on oeis.org

1, 0, 1, 1, 1, 1, 2, 4, 2, 1, 9, 15, 9, 3, 1, 44, 76, 44, 16, 4, 1, 265, 455, 265, 95, 25, 5, 1, 1854, 3186, 1854, 666, 174, 36, 6, 1, 14833, 25487, 14833, 5327, 1393, 287, 49, 7, 1, 133496, 229384, 133496, 47944, 12536, 2584, 440, 64, 8, 1, 1334961, 2293839, 1334961, 479439, 125361, 25839, 4401, 639, 81, 9, 1

Offset: 0

Werner Schulte, Apr 04 2022

Conjecture 1: T(n,k) = Sum_{i=0..n-k} (-1)^(n+k+i) * A326326(n-k,i) * n^i for 0 <= k <= n.

Conjecture 2: T(n,k) = T(n-k,0) + Sum_{i=1..n-k} T(n-k,i) * T(i+k,k) * k / (i + k - 1) for 0 < k <= n.

			The triangle T(n,k) for 0 <= k <= n starts:
n\k :       0       1       2      3      4     5    6   7  8  9
================================================================
  0 :       1
  1 :       0       1
  2 :       1       1       1
  3 :       2       4       2      1
  4 :       9      15       9      3      1
  5 :      44      76      44     16      4     1
  6 :     265     455     265     95     25     5    1
  7 :    1854    3186    1854    666    174    36    6   1
  8 :   14833   25487   14833   5327   1393   287   49   7  1
  9 :  133496  229384  133496  47944  12536  2584  440  64  8  1
  etc.

Cf. A000166 (column 0 and 2), A002467 (column 1), A006347 (column 3), A006348 (column 4), A009179 (row sums, signed), A352988 (matrix inverse).

Cf. A094587, A097807, A326326.

T := proc(n,k) option remember;
if k > n then 0 else n * T(n-1,k) + (-1)^(n-k) fi end:
for n from 0 to 9 do seq(T(n, k), k = 0..n) od; # Peter Luschny, Apr 11 2022

T(n,n) = 1 for n >= 0.
T(n,n-1) = n - 1 for n > 0.
T(n,n-2) = (n - 1)^2 for n > 1.
T(n,0) = A000166(n) for n >= 0.
T(n,1) = A002467(n) for n > 0.
T(n,2) = A000166(n) for n > 1.
T(n,k) + T(n,k+1) = (n!) / (k!) for 0 <= k <= n.
T(n,k) = (n - 1) * (T(n-1,k) + T(n-2,k)) for 0 <= k < n-1.
T(n,k) = (T(n,k-2) - (k - 2) * T(n,k-1)) / (k - 1) for 1 < k <= n.
The row polynomials p(n,x) = Sum_{k=0..n} T(n,k) * x^k satisfy recurrence equation p(n,x) = (n - 1) * (p(n-1,x) + p(n-2,x)) + x^n for n > 0 with initial value p(0,x) = 1.
Row sums are p(n,1) = abs(A009179(n)) for n >= 0.
Alternating row sums are p(n,-1) = (-1)^n for n >= 0.
T(n,k) * T(n+1,k+1) - T(n+1,k) * T(n,k+1) = (-1)^(n-k) * A094587(n,k) for 0 <= k <= n.
Define 3x3-matrices T(i,j) with n <= i <= n+2 and k <= j <= k+2. Then we have: det(T(i,j)) = 0^(n-k) for 0 <= k <= n.
E.g.f. of column k >= 0: Sum_{n>=k} T(n,k) * t^n / (n!) = (Sum_{n>=k} (-t)^n / (n!)) * (-1)^k / (1 - t).
E.g.f.: Sum_{n>=0, k=0..n} T(n,k) * x^k * t^n / (n!) = (x * exp(x * t) + exp(-t)) / ((1 + x) * (1 - t)).
p(n,x) = Sum_{k=0..n} ((n!)/(k!))*(x^(k+1) + (-1)^k)/(x + 1) for n >= 0.
T(n,k) = Sum_{i=0..n-k} (-1)^i * (n!) / ((k+i)!) for 0 <= k <= n.
T(n,k) equals matrix product of A094587 and A097807.

A352988 Matrix inverse of triangle A352650.

Original entry on oeis.org

1, 0, 1, -1, -1, 1, 0, -2, -2, 1, 0, 0, -3, -3, 1, 0, 0, 0, -4, -4, 1, 0, 0, 0, 0, -5, -5, 1, 0, 0, 0, 0, 0, -6, -6, 1, 0, 0, 0, 0, 0, 0, -7, -7, 1, 0, 0, 0, 0, 0, 0, 0, -8, -8, 1, 0, 0, 0, 0, 0, 0, 0, 0, -9, -9, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, -10, -10, 1

Offset: 0

Werner Schulte, Apr 13 2022

			The triangle T(n,k) for 0 <= k <= n starts:
n\k :   0    1    2    3    4    5    6    7    8    9
======================================================
  0 :   1
  1 :   0    1
  2 :  -1   -1    1
  3 :   0   -2   -2    1
  4 :   0    0   -3   -3    1
  5 :   0    0    0   -4   -4    1
  6 :   0    0    0    0   -5   -5    1
  7 :   0    0    0    0    0   -6   -6    1
  8 :   0    0    0    0    0    0   -7   -7    1
  9 :   0    0    0    0    0    0    0   -8   -8    1
  etc.

Cf. A094587, A097806, A352650.

T(n,n) = 1 for n >= 0, and T(n,n-1) = 1 - n for n > 0, and T(n,n-2) = 1 - n for n > 1, and T(n,k) = 0 if n < 0 or k < 0 or n < k or n > k+2.
G.f.: Sum_{n>=0, k=0..n} T(n,k) * x^k * t^n = (1 + t) * (1 - (1 + x) * t) / (1 - x * t)^2.
Alt. row sums equal (-1)^n for n >= 0.
Matrix product with A094587 yields A097806.

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.

A367962 Triangle read by rows. T(n, k) = Sum_{j=0..k} (n!/j!).

Original entry on oeis.org

1, 1, 2, 2, 4, 5, 6, 12, 15, 16, 24, 48, 60, 64, 65, 120, 240, 300, 320, 325, 326, 720, 1440, 1800, 1920, 1950, 1956, 1957, 5040, 10080, 12600, 13440, 13650, 13692, 13699, 13700, 40320, 80640, 100800, 107520, 109200, 109536, 109592, 109600, 109601

Offset: 0

Peter Luschny, Dec 06 2023

			  [0]   1;
  [1]   1,    2;
  [2]   2,    4,    5;
  [3]   6,   12,   15,   16;
  [4]  24,   48,   60,   64,   65;
  [5] 120,  240,  300,  320,  325,  326;
  [6] 720, 1440, 1800, 1920, 1950, 1956, 1957;

Cf. A094587, A000142 (T(n, 0)), A052849 (T(n, 1)), A000522 (T(n, n)), A007526 (T(n,n-1)), A038154 (T(n, n-2)), A355268 (T(n, n/2)), A367963(n) (T(2*n, n)/n!).

Cf. A001339 (row sums), A087208 (alternating row sums), A082030 (accumulated sums), A053482, A331689.

T := (n, k) -> add(n!/j!, j = 0..k):
seq(seq(T(n, k), k = 0..n), n = 0..9);

Module[{n=1},NestList[Append[n#,1+Last[#]n++]&,{1},10]] (* or *)
Table[Sum[n!/j!,{j,0,k}],{n,0,10},{k,0,n}] (* Paolo Xausa, Dec 07 2023 *)

from functools import cache
@cache
def a_row(n: int) -> list[int]:
    if n == 0: return [1]
    row = a_row(n - 1) + [0]
    for k in range(n): row[k] *= n
    row[n] = row[n - 1] + 1
    return row

def T(n, k): return sum(falling_factorial(n, n - j) for j in range(k + 1))
for n in range(9): print([T(n, k) for k in range(n + 1)])

T(n, k) = A094587(n, k) * A000522(k).
T(n, k) = e * (n! / k!) * Gamma(k + 1, 1).
Sum_{k=0..n} T(n, k) * 2^(n - k) = A053482(n).
Sum_{k=0..n} T(n, k) * binomial(n, k) = A331689(n).
Recurrence: T(n, n) = T(n, n-1) + 1 starting with T(0, 0) = 1.
For k <> n: T(n, k) = n * T(n-1, k).

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.

A138108 A triangular sequence of coefficients based on the expansion of an Hamiltonian resolvent or Green's function: p(x,t)=Exp[x*t]/(x-t); where t is taken as the Hamiltonian variable and x as the complex variable.

Original entry on oeis.org

1, 1, 0, 1, 2, 0, 2, 0, 1, 6, 0, 6, 0, 3, 0, 1, 24, 0, 24, 0, 12, 0, 4, 0, 1, 120, 0, 120, 0, 60, 0, 20, 0, 5, 0, 1, 720, 0, 720, 0, 360, 0, 120, 0, 30, 0, 6, 0, 1, 5040, 0, 5040, 0, 2520, 0, 840, 0, 210, 0, 42, 0, 7, 0, 1, 40320, 0, 40320, 0, 20160, 0, 6720, 0, 1680, 0, 336, 0, 56, 0, 8

Offset: 1

Roger L. Bagula, May 03 2008

Row sums are:
{1, 2, 5, 16, 65, 326, 1957, 13700, 109601, 986410, 9864101};
If you use a transform of;
x->Sqrt[y];
you get A094587.
The wave function form of the Green's function is:
G(x)*Phi[x,n]=Phi[x,n]/(x-E(n)).

			{1},
{1, 0, 1},
{2, 0, 2, 0, 1},
{6, 0, 6, 0, 3, 0, 1},
{24, 0, 24, 0, 12, 0, 4, 0, 1},
{120, 0, 120, 0, 60, 0, 20, 0, 5, 0, 1},
{720, 0, 720, 0, 360, 0, 120, 0, 30, 0, 6, 0, 1},
{5040, 0, 5040, 0, 2520, 0, 840, 0, 210, 0, 42, 0, 7, 0, 1},
{40320, 0, 40320, 0, 20160, 0, 6720, 0, 1680, 0, 336, 0, 56, 0, 8, 0, 1}, {362880, 0, 362880, 0, 181440, 0, 60480, 0, 15120, 0, 3024, 0, 504, 0, 72, 0, 9, 0, 1},
{3628800, 0, 3628800, 0, 1814400, 0, 604800, 0, 151200, 0, 30240, 0, 5040, 0, 720, 0, 90, 0, 10, 0, 1}

A. Messiah, Quantum mechanics, vol. 2, p. 712, fig.XVIII.2, North Holland, 1969.

Cf. A000522, A094587.

p[t_] = Exp[x*t]/(x - t); Table[ ExpandAll[x^(n + 1)*n!*SeriesCoefficient[ Series[p[t], {t, 0, 30}], n]], {n, 0, 10}]; a = Table[n!* CoefficientList[( x^(n + 1)*SeriesCoefficient[ Series[p[t], {t, 0, 30}], n]), x], {n, 0, 10}]; Flatten[a]

p(x,t)=Exp[x*t]/(x-t)=sum(P(x,n)*t^n/n!,{n,0,Infinity}); Out_n,m=n!Coefficients(x^(n+1)*P(x,n))

A143080 Triangular sequence of coefficients from an exponential based polynomial: p(x,n)=If[n == 0, 1, -(2n - 1)!x^(2n)(Sum[x^i/(i + 2n)!, {i, 0, Infinity}] - Exp[x]/x^(2n))].

Original entry on oeis.org

1, 1, 1, 6, 6, 3, 1, 120, 120, 60, 20, 5, 1, 5040, 5040, 2520, 840, 210, 42, 7, 1, 362880, 362880, 181440, 60480, 15120, 3024, 504, 72, 9, 1, 39916800, 39916800, 19958400, 6652800, 1663200, 332640, 55440, 7920, 990, 110, 11, 1, 6227020800, 6227020800

Offset: 1

Roger L. Bagula and Gary W. Adamson, Oct 15 2008

Row sums are: 1, 2, 16, 326, 13700, 986410, 108505112, ...
These polynomials are based on: f(x)=1/(1-x)-exp(x).
The n-th row is the coefficient list of the permanental polynomial of the (2n-1)X(2n-1) matrix consisting entirely of 1's (see latter Mathematica code below). - John M. Campbell, Jul 05 2012

			{1},
{1, 1},
{6, 6, 3, 1},
{120, 120, 60, 20, 5, 1},
{5040, 5040, 2520, 840, 210, 42, 7, 1},
{362880, 362880, 181440, 60480, 15120, 3024, 504, 72, 9, 1},
{39916800, 39916800, 19958400, 6652800, 1663200, 332640, 55440, 7920, 990, 110, 11, 1},
{6227020800, 6227020800, 3113510400, 1037836800, 259459200, 51891840, 8648640, 1235520, 154440, 17160, 1716, 156, 13, 1}

Cf. A000522, A094587.

Clear[f, x, n, a]; f[x_, n_] := f[x, n] = If[n == 0, 1, -(2*n - 1)!*x^(2*n)*(Sum[x^i/(i + 2*n)!, {i, 0, Infinity}] - Exp[x]/x^(2*n))]; a = Table[CoefficientList[FullSimplify[f[x, n]], x], {n, 0, 10}]; Flatten[a]
Permanent[m_List] := With[{v = Array[x, Length[m]]}, Coefficient[Times @@ (m.v), Times @@ v]]; A[q_] := Array[1 &, {q, q}]; Flatten[Table[Abs[CoefficientList[Expand[Permanent[A[2*n-1] - IdentityMatrix[2*n-1]*x]], x]], {n, 6}]] (* John M. Campbell, Jul 05 2012 *)

p(x,n)=If[n == 0, 1, -(2*n - 1)!*x^(2*n)*(Sum[x^i/(i + 2*n)!, {i, 0, Infinity}] - Exp[x]/x^(2*n))]; t(n,m)=Coefficients(p)x,n)).

A373168 Triangle read by rows: the exponential almost-Riordan array ( exp(x/(1-x)) | 1/(1-x), x ).

Original entry on oeis.org

1, 1, 1, 3, 1, 1, 13, 2, 2, 1, 73, 6, 6, 3, 1, 501, 24, 24, 12, 4, 1, 4051, 120, 120, 60, 20, 5, 1, 37633, 720, 720, 360, 120, 30, 6, 1, 394353, 5040, 5040, 2520, 840, 210, 42, 7, 1, 4596553, 40320, 40320, 20160, 6720, 1680, 336, 56, 8, 1, 58941091, 362880, 362880, 181440, 60480, 15120, 3024, 504, 72, 9, 1

Offset: 0

Stefano Spezia, May 26 2024

			The triangle begins:
     1;
     1,   1;
     3,   1,   1;
    13,   2,   2,  1;
    73,   6,   6,  3,  1;
   501,  24,  24, 12,  4, 1;
  4051, 120, 120, 60, 20, 5, 1;
  ...

Y. Alp and E. G. Kocer, Exponential Almost-Riordan Arrays, Results Math 79, 173 (2024). See page 13.

Cf. A000142, A000262 (k=0), A001710, A001715, A001720, A001725, A001730, A049388, A049389, A049398, A051431.

Triangle A094587 with 1st column A000262.

T[n_,0]:=n!SeriesCoefficient[Exp[x/(1-x)],{x,0,n}]; T[n_,k_]:=(n-1)!/(k-1)!SeriesCoefficient[1/(1-x)*x^(k-1),{x,0,n-1}]; Table[T[n,k],{n,0,10},{k,0,n}]//Flatten

T(n,0) = n! * [x^n] exp(x/(1-x)); T(n,k) = (n-1)!/(k-1)! * [x^(n-1)] 1/(1-x)*x^(k-1).
T(n,3) = A001710(n-1) for n > 2.
T(n,4) = A001715(n-1) for n > 3.
T(n,5) = A001720(n-1) for n > 4.
T(n,6) = A001725(n-1) for n > 5.
T(n,7) = A001730(n-1) for n > 6.
T(n,8) = A049388(n-8) for n > 7.
T(n,9) = A049389(n-9) for n > 8.
T(n,10) = A049398(n-10) for n > 9.
T(n,11) = A051431(n-11) for n > 10.

Previous Showing 41-50 of 51 results. Next

cp's OEIS Frontend

A070895 Triangle read by rows where T(n+1,k)=T(n,k)+n*T(n-1,k) starting with T(n,n)=1 and T(n,k)=0 if n

Views

Author

Keywords

Comments

Examples

Crossrefs

Formula

A348482 Triangle read by rows: T(n,k) = (Sum_{i=k..n} i!)/(k!) for 0 <= k <= n.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

Formula

A352650 Triangle read by rows: T(n,k) = n * T(n-1,k) + (-1)^(n-k) for 0 <= k <= n with initial values T(n,k) = 0 if n < 0 or k < 0 or k > n.

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

Maple

Formula

A352988 Matrix inverse of triangle A352650.

Views

Author

Keywords

Examples

Crossrefs

Formula

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

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

PARI

Formula

A367962 Triangle read by rows. T(n, k) = Sum_{j=0..k} (n!/j!).

Views

Author

Keywords

Examples

Crossrefs

Programs

Maple

Mathematica

Python

SageMath

Formula

A376582 Triangle of generalized Stirling numbers.

Views

Author

Keywords

Examples

Crossrefs

Programs

Maple

Formula

A138108 A triangular sequence of coefficients based on the expansion of an Hamiltonian resolvent or Green's function: p(x,t)=Exp[x*t]/(x-t); where t is taken as the Hamiltonian variable and x as the complex variable.

Views

Author

Keywords

Comments

Examples

References

Crossrefs

Programs

Mathematica

Formula

A143080 Triangular sequence of coefficients from an exponential based polynomial: p(x,n)=If[n == 0, 1, -(2n - 1)!x^(2n)(Sum[x^i/(i + 2n)!, {i, 0, Infinity}] - Exp[x]/x^(2n))].