A088956 -id:A088956 - cp's OEIS frontend

A089468 Hyperbinomial transform of A089467 and also the 2nd hyperbinomial transform of A089466.

1, 3, 15, 110, 1083, 13482, 203569, 3618540, 74058105, 1715620148, 44384718879, 1268498827752, 39692276983555, 1349678904881400, 49556966130059553, 1954156038072106448, 82363978221026323761, 3695194039210436996400

Offset: 0

Paul D. Hanna, Nov 08 2003

nonn

See A088956 for the definition of the hyperbinomial transform.

Cf. A089466, A089467, A088956.

CoefficientList[Series[(LambertW[-x]^2*E^(-1/2*LambertW[-x]^2))/(x^2*(1+LambertW[-x])), {x, 0, 20}], x]* Range[0, 20]! (* Vaclav Kotesovec, Jul 09 2013 *)

a(n)=if(n<0,0,sum(m=0,n,sum(j=0,m,binomial(m,j)*binomial(n,n-m-j)*(n+1)^(n-m-j)*(m+j)!/(-2)^j)/m!))

a(n) = Sum_{k=0..n} (n-k+1)^(n-k-1)*C(n, k)*A089467(k).
a(n) = Sum_{k=0..n} 2*(n-k+2)^(n-k-1)*C(n, k)*A089466(k).
a(n) = Sum_{m=0..n} (Sum_{j=0..m} C(m, j)*C(n, n-m-j)*(n+1)^(n-m-j)*(m+j)!/(-2)^j)/m!.
E.g.f.: (LambertW(-x)^2*exp(-1/2*LambertW(-x)^2))/(x^2*(1+LambertW(-x))). - Vladeta Jovovic, Oct 26 2004
a(n) ~ exp(3/2)*n^n. - Vaclav Kotesovec, Jul 09 2013

A218502 10th iteration of the hyperbinomial transform on the sequence of 1's.

Original entry on oeis.org

1, 11, 141, 2081, 34961, 661601, 13970521, 326429401, 8377176001, 234573153281, 7125155956601, 233554674134441, 8223284332647361, 309711995280132001, 12430859603012736601, 529915231307371964201, 23918971999180778999681, 1139982481554110359552001

Offset: 0

Alois P. Heinz, Oct 30 2012

nonn

See A088956 for the definition of the hyperbinomial transform.

Alois P. Heinz, Table of n, a(n) for n = 0..150

Column k=10 of A144303.

a:= n-> add(10*(n-j+10)^(n-j-1)*binomial(n,j), j=0..n):
seq (a(n), n=0..20);

Table[Sum[10*(n-j+10)^(n-j-1)*Binomial[n,j],{j,0,n}],{n,0,20}] (* Vaclav Kotesovec, Oct 18 2013 *)

E.g.f.: exp(x) * (-LambertW(-x)/x)^10.
a(n) = Sum_{j=0..n} 10 * (n-j+10)^(n-j-1) * C(n,j).
Hyperbinomial transform of A218501.
a(n) ~ 10*exp(10+exp(-1))*n^(n-1). - Vaclav Kotesovec, Oct 18 2013

A227325 a(n) = A000272(n+1) * A000984(n).

Original entry on oeis.org

1, 2, 18, 320, 8750, 326592, 15529668, 899678208, 61556811030, 4862000000000, 435644983598396, 43678490079264768, 4846281282497517772, 589650705050503577600, 78074729079345703125000, 11177395284330167371038720, 1720546364146510165684599270

Offset: 0

Reinhard Zumkeller, Jul 07 2013

nonn

Reinhard Zumkeller, Table of n, a(n) for n = 0..250

Cf. A000272, A000984.

Central terms of triangle A088956.

Haskell
```
a227325 n = a000272 (n + 1) * a000984 n
```

a(n) = A088956(2*n, n) = (n+1)^(n-1) * C(2*n, n).

Typo in formula fixed by Zak Seidov, Jul 08 2013

A089470 Self-convolution of this sequence is equal to its hyperbinomial transform and results in A089471.

Original entry on oeis.org

1, 1, 4, 29, 303, 4108, 68165, 1334403, 30056112, 764920823, 21694511367, 678288426792, 23173084581845, 858785085529061, 34311202499100416, 1470080434980994825, 67236889676684657943, 3269565144147886318168

Offset: 0

Paul D. Hanna, Nov 07 2003

nonn

See A088956 for the definition of the hyperbinomial transform.

			The self-convolution at n=4: 303*1+29*1+4*4+1*29+1*303 = 680 = A089471(4) and equals the hyperbinomial transform at n=4: 125*1+64*1+18*4+4*29+1*303 = 680 = A089471(4).

Cf. A089471, A088956.

A089471(n) = sum(k=1, n, a(k)*a(n-k)); A089471(n) = sum(k=0, n, (n-k+1)^(n-k-1)*binomial(n, k)*a(k)).

A089471 Equals the self-convolution of A089470 and also the hyperbinomial transform of A089470.

Original entry on oeis.org

1, 2, 9, 66, 680, 9054, 147811, 2855574, 63656423, 1607072112, 45294892304, 1409197189256, 47954491442089, 1771493331491354, 70590010219153189, 3017771375030039066, 137763757493141082536, 6688261925293875095950

Offset: 0

Paul D. Hanna, Nov 07 2003

nonn

See A088956 for the definition of the hyperbinomial transform.

			The self-convolution of A089470 at n=4: a(4) = 680 = 303*1+29*1+4*4+1*29+1*303 and equals the hyperbinomial transform of A089470 at n=4: a(4) = 680 = 125*1+64*1+18*4+4*29+1*303.

Cf. A089470, A088956.

a(n) = sum(k=1, n, A089470(k)*A089470(n-k)); a(n) = sum(k=0, n, (n-k+1)^(n-k-1)*binomial(n, k)*A089470(k)).

A203092 Triangular array read by rows. T(n,k) is the number of partial functions on {1,2,...,n} that are endofunctions with no cycles of length > 1 that have exactly k components.

Original entry on oeis.org

1, 1, 1, 1, 4, 1, 1, 18, 9, 1, 1, 116, 78, 16, 1, 1, 1060, 810, 220, 25, 1, 1, 12702, 10335, 3260, 495, 36, 1, 1, 187810, 158613, 54740, 9835, 966, 49, 1, 1, 3296120, 2854908, 1046024, 209510, 24696, 1708, 64, 1

Offset: 0

Geoffrey Critzer, Dec 29 2011

Row sums = A088957.
T(n,0)= 1,  the empty function.
T(n,n)= 1,  the identity function.
T(n,n-1)= n^2  (apparently).

			T(2,1)= 4 because there are 4 such partial functions on {1,2}: 1->1, 2->2, 1->1 2->1, 1->2 2->2,
1
1     1
1     4     1
1     18    9     1
1     116   78    16    1
1     1060  810   220   25    1
1     12702 10335 3260  495   36    1

Cf. A088956, A144289

nn = 8; t = Sum[n^(n - 1) x^n/n!, {n, 1, nn}];
f[list_] := Select[list, # > 0 &];
Map[f, Range[0, nn]! CoefficientList[ Series[Exp[x] Exp[y t], {x, 0, nn}], {x, y}]] // Flatten

E.g.f.: exp(x)*exp(y T(x)) where T(x) is the e.g.f. for A000169.

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.

A089459 Shifts 2 places left under the hyperbinomial transform.

Original entry on oeis.org

1, 1, 1, 2, 6, 30, 221, 2201, 27769, 424757, 7639190, 157969167, 3692733181, 96293669499, 2771232779481, 87247362424120, 2982901522141490, 110057420712635526, 4358745145279372293, 184433423940319376323

Offset: 0

Paul D. Hanna, Nov 04 2003

nonn

Cf. A088956, A001858.

a(n+2) = sum(k=0, n, (n-k+1)^(n-k-1)*binomial(n, k)*a(k)), for n>=0, with a(0)=1, a(1)=1.

A089469 a(n+1) = the n-th term of the n-th binomial transform.

Original entry on oeis.org

1, 1, 2, 10, 82, 946, 14246, 267974, 6117202, 166015698, 5273053710, 193534712510, 8119820921626, 385777848702394, 20583872009571798, 1224407374239009622, 80669343513439179922, 5852864801437926734482, 465237079520383362585598

Offset: 0

Paul D. Hanna, Nov 08 2003

nonn

Form a square array where the n-th row is the n-th binomial transform of this sequence, starting with this sequence in the zeroth row; then the diagonal of the square array so formed is this sequence shifted 1 place left.

			Note the diagonal in the array of iterated binomial transforms:
[_1,1,2,10,82,946,14246,267974,..]
[1,_2,5,20,139,1482,21389,390832,..]
[1,3,_10,42,258,2438,32854,577362,..]
[1,4,17,_82,499,4264,52361,869270,..]
[1,5,26,146,_946,7770,87350,1346062,..]
[1,6,37,240,1707,_14246,151501,2159484,..]
[1,7,50,370,2914,25582,_267974,3588122,..]
[1,8,65,542,4723,44388,473369,_6117202,..]

Cf. A071207, A088956.

{L=20; a=[1]; for(i=1,L,b=a; for(n=0,length(a)-1, b[n+1]=sum(k=0,n,a[k+1]*binomial(n,k)*n^(n-k)); ); a=concat(1,b); ); for(j=1,L,print1(a[j],","))}

a(n+1) = sum(k=0, n, a(k)*binomial(n, k)*n^(n-k))

cp's OEIS Frontend

A089468 Hyperbinomial transform of A089467 and also the 2nd hyperbinomial transform of A089466.

Views

Author

Keywords

Comments

Crossrefs

Programs

Mathematica

PARI

Formula

A218502 10th iteration of the hyperbinomial transform on the sequence of 1's.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Maple

Mathematica

Formula

A227325 a(n) = A000272(n+1) * A000984(n).

Views

Author

Keywords

Links

Crossrefs

Programs

Haskell

Formula

Extensions

A089470 Self-convolution of this sequence is equal to its hyperbinomial transform and results in A089471.

Views

Author

Keywords

Comments

Examples

Crossrefs

Formula

A089471 Equals the self-convolution of A089470 and also the hyperbinomial transform of A089470.

Views

Author

Keywords

Comments

Examples

Crossrefs

Formula

A203092 Triangular array read by rows. T(n,k) is the number of partial functions on {1,2,...,n} that are endofunctions with no cycles of length > 1 that have exactly k components.

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

Mathematica

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

A089459 Shifts 2 places left under the hyperbinomial transform.

Views

Author

Keywords

Crossrefs

Formula

A089469 a(n+1) = the n-th term of the n-th binomial transform.

Views

Author

Keywords

Comments