A111577 -id:A111577 - cp's OEIS frontend

A290596 Triangle read by rows. A generalization of unsigned Lah numbers, called L[3,1].

1, 2, 1, 10, 10, 1, 80, 120, 24, 1, 880, 1760, 528, 44, 1, 12320, 30800, 12320, 1540, 70, 1, 209440, 628320, 314160, 52360, 3570, 102, 1, 4188800, 14660800, 8796480, 1832600, 166600, 7140, 140, 1, 96342400, 385369600, 269758720, 67439680, 7663600, 437920, 12880, 184, 1, 2504902400, 11272060800, 9017648640, 2630147520, 358656480, 25618320, 1004640, 21528, 234, 1, 72642169600, 363210848000, 326889763200, 108963254400, 17335063200, 1485862560, 72836400, 2081040, 33930, 290, 1

Offset: 0

Wolfdieter Lang, Sep 13 2017

For the general L[d,a] triangles see A286724, also for references.
This is the generalized signless Lah number triangle L[3,1], the Sheffer triangle ((1 - 3*t)^(-2/3), t/(1 - 3*t)). It is defined as transition matrix
risefac[3,1](x, n) = Sum_{m=0..n} L[3,1](n, m)*fallfac[3,1](x, m), where risefac[3,1](x, n):= Product_{0..n-1} (x  + (1 + 3*j)) for n >= 1 and risefac[3,1](x, 0) := 1, and  fallfac[3,1](x, n):= Product_{0..n-1} (x - (1 + 3*j)) for n >= 1 and fallfac[3,1](x, 0) := 1.
In matrix notation: L[3,1] = S1phat[3,1]*S2hat[3,1] with the unsigned scaled Stirling1 and the scaled Stirling2 generalizations A286718 and A111577 (but here with offsets 0), respectively.
The a- and z-sequences for this Sheffer matrix has e.g.f.s Ea(t) = 1 + 3*t and (Ez(t) = (1 + 3*t)*(1 - (1 + 3*t)^(-2/3))/t, respectively. That is, a = {1, 3, repeat(0)} and z(n) = A290597(n)/A038500(n+1). For the proof see the second W. Lang link. See also a W. Lang link under A006232 for Sheffer a- and z-sequences with references (in the Riordan case).
The inverse matrix T^(-1) = L^(-1)[3,1] is Sheffer ((1 + 3*t)^(-2/3), t/(1 + 3*t)). This means that  T^(-1)(n, m) = (-1)^(n-m)*T(n, m).
fallfac[3,1](x, n) = Sum_{m=0..n} (-1)^(n-m)*T(n, m)*risefac[3,1](x, m), n >= 0.

			The triangle T(n, m) begins:
n\m         0         1         2        3       4      5     6   7 8  ...
0:          1
1:          2         1
2:         10        10         1
3:         80       120        24        1
4:        880      1760       528       44       1
5:      12320     30800     12320     1540      70      1
6:     209440    628320    314160    52360    3570    102     1
7:    4188800  14660800   8796480  1832600  166600   7140   140   1
8:   96342400 385369600 269758720 67439680 7663600 437920 12880 184 1
...
n = 9: 2504902400 11272060800 9017648640 2630147520 358656480 25618320 1004640 21528 234 1,
n = 10: 72642169600 363210848000 326889763200 108963254400 17335063200 1485862560 72836400 2081040 33930 290 1.
...
Recurrence from a-sequence:  T(4, 2) = 2*T(3, 1) + 3*4*T(3, 2) = 2*120 + 12*24 = 528.
Recurrence from z-sequence: T(4, 0) = 4*(z(0)*T(3, 0) + z(1)*T(3, 1) + z(2)*T(3, 2) + z(3)*T(3, 3)) = 4*(2*80 + 1*120 - (10/3)*24 + 20*1) = 880.
Four term recurrence: T(4, 2) = T(3, 1) + 2*10*T(3, 2) - 3*3*8*T(2, 2) =  120 + 20*24 - 72*1 = 528.
Meixner type identity for n = 2: (D_x - 3*(D_x)^2)*(10 + 10*x + x^2 ) = (10 + 2*x) - 3*2 = 2*(2 + x).
Sheffer recurrence for R(3, x): [(2 + x) + 6*(1 + x)*D_x + 9*x*(D_x)^2] (10 + 10*x + x^2) = (2 + x)*(10 + 10*x + x^2) + 6*(1 + x)*(10 +2*x) + 9*2*x = 80 + 120*x + 24*x^2 + x^3 = R(3, x).
Boas-Buck recurrence for column m = 2 with n = 4: T(4, 2) = (4!*8/2)*(1*24/3! + 3*1/2!) = 528.

Steven Roman, The Umbral Calculus, Academic press, Orlando, London, 1984, p. 50.

Wolfdieter Lang, On Sums of Powers of Arithmetic Progressions, and Generalized Stirling, Eulerian and Bernoulli Numbers, arXiv:math/1707.04451 [math.NT], July 2017.
Wolfdieter Lang, Note on a- and z-sequences of Sheffer number triangles for certain generalized Lah numbers.

Cf. A008544 (column m=0), A038500, A111577, A271703 L[1,0], A286718, A286724 L[2,1], A290597, A290598 L[3,2].

T(n, m) = L[3,1](n,m) = Sum_{k=m..n} A286718(n, k)*A111577(k+1, m+1), 0 <= m <= n.
E.g.f. of row polynomials R(n, x) := Sum_{m=0..n} T(n, m)*x^m:
(1 - 3*t)^(-2/3)*exp(x*t/(1 - 3*t)) (this is the e.g.f. for the triangle).
E.g.f. of column m: (1 - 3*t)^(-2/3)*(t/(1 - 3*t))^m/m!, m >= 0.
Three term recurrence for column entries m >= 1: T(n, m) = (n/m)*T(n-1, m-1) + 3*n*T(n-1, m) with T(n, m) = 0 for n < m, and for the column m = 0: T(n, 0) = n*Sum_{j=0}^(n-1) z(j)*T(n-1, j), from the a-sequence {1, 3 repeat(0)} and the z-sequence given above.
Four term recurrence: T(n, m) = T(n-1, m-1) + 2*(3*n - 2)*T(n-1, m) - 3*(n-1)*(3*n - 4)*T(n-2, m), n >= m >= 0, with T(0, 0) = 1, T(-1, m) = 0, T(n, -1) = 0 and T(n, m) = 0 if n < m.
Meixner type identity for (monic) row polynomials: (D_x/(1 + 3*D_x)) * R(n, x) = n*R(n-1, x), n >= 1, with R(0, x) = 1 and D_x = d/dx. That is, Sum_{k=0..n-1} (-3)^k*(D_x)^(k+1)*R(n, x) = n*R(n-1, x), n >= 1.
General recurrence for Sheffer row polynomials (see the Roman reference, p. 50, Corollary 3.7.2, rewritten for the present Sheffer notation):
R(n, x) = [(2 + x)*1 + 6*(1 + x)*D_x + 3^2*x*(D_x)^2]*R(n-1, x), n >= 1, with R(0, x) = 1.
Boas-Buck recurrence for column m (see a comment in A286724 with references): T(n, m) = (n!/(n-m))*(2 + 3*m)*Sum_{p=0..n-1-m} 3^p*T(n-1-p, m)/(n-1-p)!, for n > m >= 0, with input T(m, m) = 1.

A021874 Expansion of 1/((1-x) * (1-4x) (1-7x) (1-10*x)).

Original entry on oeis.org

1, 22, 325, 4070, 46781, 511742, 5430405, 56516790, 580744461, 5916830062, 59935396885, 604729235110, 6084941584541, 61113049957982, 612976296281765, 6142684971387030, 61517309500479021, 615806336417543502, 6162496145690677045, 61655991294017340550, 616777123265962899901

Offset: 0

N. J. A. Sloane

Vincenzo Librandi, Table of n, a(n) for n = 0..200
Index entries for linear recurrences with constant coefficients, signature (22,-159,418,-280).

Cf. A002450, A016223, A111577.

Cf. A000012, A282629.

CoefficientList[Series[1 / ((1 - x) (1 - 4 x) (1 - 7 x) (1 - 10 x)), {x, 0, 20}], x] (* Vincenzo Librandi, Jul 11 2013 *)
LinearRecurrence[{22,-159,418,-280},{1,22,325,4070},30] (* Harvey P. Dale, May 13 2018 *)

a(n) = (10^(n+3) - 3*7^(n+3) + 3*4^(n+3) - 1)/162. - Yahia Kahloune, Jul 05 2013
E.g.f.: exp(x)*(1000*exp(9*x) - 1029*exp(6*x) + 192*exp(3*x) - 1)/(3!*3^3). This is d^3/dx^3 exp(x)*(exp(3*x - 1))^3/(3!*3^3); see also column m=3 of A282629 divided by 3^3. The o.g.f. is given in the name. - Wolfdieter Lang, Apr 08 2017
a(n) = Sum_{k=0..n} 3^k * binomial(n+3,k+3) * Stirling2(k+3,3). - Seiichi Manyama, May 03 2025

A111669 Triangle read by rows, based on a simple Fibonacci recursion rule.

Original entry on oeis.org

1, 1, 1, 1, 2, 1, 1, 3, 4, 1, 1, 4, 11, 7, 1, 1, 5, 26, 32, 12, 1, 1, 6, 57, 122, 92, 20, 1, 1, 7, 120, 423, 582, 252, 33, 1, 1, 8, 247, 1389, 3333, 2598, 681, 54, 1, 1, 9, 502, 4414, 18054, 24117, 11451, 1815, 88, 1, 1, 10, 1013, 13744, 94684, 210990, 172980, 49566, 4807, 143, 1

Offset: 0

Gary W. Adamson, Aug 14 2005

Subdiagonal is A000071(n+3). Row sums of inverse are 0^n.

Row sums are given by A135934. - Emanuele Munarini, Dec 05 2017

			Triangle begins
  1....1....2....3....5....8...13....F(k+1)
  1
  1....1
  1....2....1
  1....3....4....1
  1....4...11....7....1
  1....5...26...32...12....1
  1....6...57..122...92...20....1
For example, T(6,3) = 122 = 26 + 3*32 = T(5,2) + F(4)*T(5,3).

Michel Marcus, Rows n = 0..20, flattened

Cf. A111577, A111578, A111579, A008277, A039755, A135934.

(* To generate the triangle *)
Grid[RecurrenceTable[{F[n,k] == F[n-1,k-1] + Fibonacci[k+1] F[n-1,k], F[0,k] == KroneckerDelta[k]}, F, {n,0,10}, {k,0,10}]] (* Emanuele Munarini, Dec 05 2017 *)

T(n, k) = if ((n<0) || (k<0), 0, if ((n==0) && (k==0), 1, T(n-1, k-1) + fibonacci(k+1)*T(n-1, k))); \\ Michel Marcus, May 25 2024

T(n, k) = T(n-1, k-1) + F(k+1)*T(n-1, k) where F(n)=A000045(n).

Column k has g.f. x^k/Product_{j=0..k} (1 - F(j+1)*x).

Edited by Paul Barry, Nov 14 2005

A111579 Triangle A(r,c) read by rows, which contains the row sums of the triangle T(n,k)= T(n-1,k-1)+((c-1)k+1)T(n-1,k) in column c.

Original entry on oeis.org

1, 1, 1, 1, 2, 1, 1, 4, 2, 1, 1, 8, 5, 2, 1, 1, 16, 15, 6, 2, 1, 1, 32, 52, 24, 7, 2, 1, 1, 64, 203, 116, 35, 8, 2, 1, 1, 128, 877, 648, 214, 48, 9, 2, 1, 1, 256, 4140, 4088, 1523, 352, 63, 10, 2, 1, 1, 512, 21147, 28640, 12349, 3008, 536, 80, 11, 2, 1

Offset: 0

Gary W. Adamson, Aug 07 2005

Triangles of generalized Stirling numbers of the second kind may be defined by recurrences T(n,k) = T(n-1,k-1) + Q*T(n-1,k) initialized by T(0,0)=T(1,0)=T(1,1)=1. Q=1 generates Pascal's triangle A007318,
Q=k+1 generates A008277, Q=2k+1 generates A039755, Q=3k+1 generates A111577, Q=4k+1 generates A111578, Q=5k+1 generates A166973.
(These definitions assume row and column enumeration 0<=n, 0<=k<=n.)
Each of these triangles characterized by Q=(c-1)*k+1 has row sums sum_{k=0..n} T(n,k), which define the column A(.,c).

Cf. A008277, A000110, A039755, A004211, A111577, A111578.

T := proc(n,k,c) if k < 0 or k > n then 0 ; elif n <= 1 then 1; else procname(n-1,k-1,c)+((c-1)*k+1)*procname(n-1,k,c) ; fi; end:
A111579 := proc(r,c) local n; if c = 0 then 1 ; else n := r-c ; add( T(n,k,c),k=0..n) ; end if; end:
seq(seq(A111579(r,c),c=0..r),r=0..10) ; # R. J. Mathar, Oct 30 2009

T[n_, k_, c_] := T[n, k, c] = If[k < 0 || k > n, 0, If[n <= 1, 1, T[n-1, k-1, c] + ((c-1)*k+1)*T[n-1, k, c]]];
A111579[r_, c_] := Module[{n}, If[c == 0, 1, n = r - c; Sum[T[n, k, c], {k, 0, n}]]];
Table[A111579[r, c], {r, 0, 10}, {c, 0, r}] // Flatten (* Jean-François Alcover, Aug 01 2023, after R. J. Mathar *)

A(r=n+c,c) = sum_{k=0..n} T(n,k,c), 0<=c<=r where T(n,k,c) = T(n-1,k-1,c) + ((c-1)*k+1)*T(n-1,k,c).
A(r,0) = 1.
A(r,1) = 2^(r-1).
A(r,2) = A000110(r-1).
A(r,3) = A007405(r-3).

Edited by R. J. Mathar, Oct 30 2009

A166973 Triangle T(n,k) read by rows: T(n, k) = (mn - mk + 1)T(n - 1, k - 1) + (5k - 4)(mk - (m - 1))*T(n - 1, k) where m = 0.

Original entry on oeis.org

1, 1, 1, 1, 7, 1, 1, 43, 18, 1, 1, 259, 241, 34, 1, 1, 1555, 2910, 785, 55, 1, 1, 9331, 33565, 15470, 1940, 81, 1, 1, 55987, 378546, 281085, 56210, 4046, 112, 1, 1, 335923, 4219993, 4875906, 1461495, 161406, 7518, 148, 1, 1, 2015539, 46755846, 82234489

Offset: 1

Roger L. Bagula, Oct 26 2009

The recursion T(n, k) = (m*n - m*k + 1)*T(n-1, k-1) + (5*k - 4)*(m*k - (m - 1))*T(n-1, k) was intended to range over m values 0 to 4 as given by the original Mathematica code. This sequences is the case for m = 0. - G. C. Greubel, May 29 2016

With offset 0 in the rows and columns this is the Sheffer triangle S2[5,1] = (exp(x), (exp(5*x) - 1)/5). See S2[4,1] = A111578 (with offsets 0), S[3,1] = A111577 (with offsets 0), S2[2,1] = A039755

			Triangle T(n, k) starts:
n\k   1       2        3        4        5       6      7     8   9 10 ...
1:    1
2:    1       1
3:    1       7        1
4:    1      43       18        1
5:    1     259      241       34        1
6:    1    1555     2910      785       55       1
7:    1    9331    33565    15470     1940      81      1
8:    1   55987   378546   281085    56210    4046    112     1
9:    1  335923  4219993  4875906  1461495  161406   7518   148   1
10:   1 2015539 46755846 82234489 35567301 5658051 394464 12846 189  1
... Reformatted, - _Wolfdieter Lang_, Aug 13 2017

G. C. Greubel, Table of n, a(n) for the first 25 rows

Cf. A111577.

S2[4,1] = A111578 (with offsets 0), S2[3,1] = A111577 (with offsets 0), S2[2,1] = A039755. - Wolfdieter Lang, Aug 13 2017

A[n_, 1] := 1; A[n_, n_] := 1; A[n_, k_] := A[n - 1, k - 1] + (5*k - 4)*A[n - 1,k]; Flatten[ Table[A[n, k], {n, 10}, {k, n}]] (* modified by G. C. Greubel, May 29 2016 *)

T(n, k) = T(n - 1, k - 1) + (5*k - 4)*T(n - 1, k).

E.g.f. column k: int(exp(x)*((exp(5*x)-1)/5)^(k-1)/(k-1)!, x) + (-1)^k/A008548(k). - Wolfdieter Lang, Aug 13 2017

A166972 Triangle T(n,k) read by rows: T(n,k) = (n-k+1)T(n-1,k-1) + (3k-2)kT(n-1,k), initialized by T(n,1) = T(n,n) = 1.

Original entry on oeis.org

1, 1, 1, 1, 10, 1, 1, 83, 41, 1, 1, 668, 1110, 122, 1, 1, 5349, 25982, 8210, 309, 1, 1, 42798, 572367, 432328, 44715, 714, 1, 1, 342391, 12276495, 20154955, 4635787, 202689, 1561, 1, 1, 2739136, 260203132, 879857170, 402100930, 38001292, 815680

Offset: 1

Roger L. Bagula, Oct 26 2009

The row sums are: 1, 2, 12, 126, 1902, 39852, 1092924, 37613880, 1583720640, 79861657752,...

The original format of this sequence used the recursion T(n,k) = (m*n-m*k+1)*T(n-1, k-1) + (3*k-2)*(m*k-(m-1))*T(n-1, k) for varying values of m. - G. C. Greubel, May 29 2016

			1;
1, 1;
1, 10, 1;
1, 83, 41, 1;
1, 668, 1110, 122, 1;
1, 5349, 25982, 8210, 309, 1;
1, 42798, 572367, 432328, 44715, 714, 1;
1, 342391, 12276495, 20154955, 4635787, 202689, 1561, 1;
1, 2739136, 260203132, 879857170, 402100930, 38001292, 815680, 3298, 1;
1, 21913097, 5486178860, 37015708724, 31415703470, 5658628682, 260490608, 3027488, 6821, 1;

G. C. Greubel, Table of n, a(n) for n = 1..325

Cf. A111577.

A166972 := proc(n,k)
        if k = 1 or k= n then
                1;
        else
                (n-k+1)*procname(n-1,k-1)+(3*k-2)*k*procname(n-1,k) ;
        end if;
end proc: # R. J. Mathar, Nov 05 2011

A[n_, 1] := 1; A[n_, n_] := 1; A[n_, k_] := (n - k + 1)*A[n - 1, k - 1] + (3*k - 2)*k*A[n - 1, k]; Flatten[ Table[A[n, k], {n, 10}, {k, n}]] (* modified by G. C. Greubel, May 29 2016 *)

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A290596 Triangle read by rows. A generalization of unsigned Lah numbers, called L[3,1].

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A021874 Expansion of 1/((1-x) * (1-4*x) * (1-7*x) * (1-10*x)).

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A111669 Triangle read by rows, based on a simple Fibonacci recursion rule.

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A111579 Triangle A(r,c) read by rows, which contains the row sums of the triangle T(n,k)= T(n-1,k-1)+((c-1)*k+1)*T(n-1,k) in column c.

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A166973 Triangle T(n,k) read by rows: T(n, k) = (m*n - m*k + 1)*T(n - 1, k - 1) + (5*k - 4)*(m*k - (m - 1))*T(n - 1, k) where m = 0.

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A166972 Triangle T(n,k) read by rows: T(n,k) = (n-k+1)*T(n-1,k-1) + (3*k-2)*k*T(n-1,k), initialized by T(n,1) = T(n,n) = 1.

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A021874 Expansion of 1/((1-x) * (1-4x) (1-7x) (1-10*x)).

A111579 Triangle A(r,c) read by rows, which contains the row sums of the triangle T(n,k)= T(n-1,k-1)+((c-1)k+1)T(n-1,k) in column c.

A166973 Triangle T(n,k) read by rows: T(n, k) = (mn - mk + 1)T(n - 1, k - 1) + (5k - 4)(mk - (m - 1))*T(n - 1, k) where m = 0.

A166972 Triangle T(n,k) read by rows: T(n,k) = (n-k+1)T(n-1,k-1) + (3k-2)kT(n-1,k), initialized by T(n,1) = T(n,n) = 1.