A000340 -id:A000340 - cp's OEIS frontend

A156923 Fourth right hand column (n-m=3) of the A156920 triangle.

1, 37, 646, 8030, 82610, 756218, 6411720, 51586344, 400011435, 3020658295, 22373863774, 163379472214, 1180488191108, 8462445970580, 60305767988960, 427848087263712, 3025286818472661

Offset: 0

Johannes W. Meijer, Feb 20 2009

Other columns A000340, A156922, A156924.
Equals A156920 fourth right hand column.
Equals A156919 fourth right hand column divided by 8.
Equals A142963 fourth right hand column divided by 2^n

a(n) = 30*a(n-1)-385*a(n-2)+2776*a(n-3)-12418*a(n-4)+35908*a(n-5)-67818*a(n-6)+82552*a(n-7)-62109*a(n-8)+26190*a(n-9)-4725*a(n-10)
a(n) = (-8*n^3+972*n^2*3^n-84*n^2+7776*n*3^n-11250*n*5^n-286*n+15309*3^n-50625*5^n+36015*7^n-315)/384
G.f.: GF1(z;RHCnr=4) = (1+7*z-79*z^2+119*z^3+126*z^4-270*z^5)/((1-7*z)*(1-5*z)^2*(1-3*z)^3*(1-z)^4)

A156924 Fifth right hand column (n-m=4) of the A156920 triangle.

Original entry on oeis.org

1, 83, 2685, 56285, 919615, 12813843, 160206627, 1854550395, 20291056470, 212826091180, 2161547322134, 21414479565774, 208076662576370, 1991164206775450, 18825064380813450

Offset: 0

Johannes W. Meijer, Feb 20 2009

Other columns A000340, A156922, A156923.
Equals A156920 fifth right hand column.
Equals A156919 fifth right hand column divided by 16.
Equals A142963 fifth right hand column divided by 2^n

a(n)=55*a(n-1)-1365*a(n-2)+20251*a(n-3)-200557*a(n-4)+1402203*a(n-5)-7137473*a(n-6)+26886431*a(n-7)-75433971*a(n-8)+157376597*a(n-9)-241846607*a(n-10)+268663713*a(n-11)-208880991*a(n-12)+107416665*a(n-13)-32730075*a(n-14)+4465125*a(n-15)
a(n)= (16*n^4-7776*n^3*3^n+256*n^3-104976*n^2*3^n+225000*n^2*5^n+ 1496*n^2- 464616*n*3^n+ 2250000*n*5^n-2016840*n*7^n+3776*n-673596*3^n+5568750*5^n-11092620*7^n+6200145*9^n+3465)/6144
G.f.: GF1(z;RHCnr=5) = (1+28*z-515*z^2+1654*z^3+8689*z^4-65864*z^5+142371*z^6-82242*z^7-99090*z^8+113400*z^9)/((1-9*z)*(1-7*z)^2*(1-5*z)^3*(1-3*z)^4*(1-z)^5)

A333143 Triangle read by rows: T(n, k) = qStirling2(n, k, q) for q = 3, with 0 <= k <= n.

Original entry on oeis.org

1, 1, 1, 1, 5, 1, 1, 21, 18, 1, 1, 85, 255, 58, 1, 1, 341, 3400, 2575, 179, 1, 1, 1365, 44541, 106400, 24234, 543, 1, 1, 5461, 580398, 4300541, 3038714, 221886, 1636, 1, 1, 21845, 7550635, 172602038, 371984935, 83805218, 2010034, 4916, 1

Offset: 0

Peter Luschny, Mar 09 2020

			[0] 1
[1] 1, 1
[2] 1, 5,     1
[3] 1, 21,    18,      1
[4] 1, 85,    255,     58,        1
[5] 1, 341,   3400,    2575,      179,       1
[6] 1, 1365,  44541,   106400,    24234,     543,      1
[7] 1, 5461,  580398,  4300541,   3038714,   221886,   1636,    1
[8] 1, 21845, 7550635, 172602038, 371984935, 83805218, 2010034, 4916, 1

T(n, 1) = A002450(n), T(n, n-1) = A000340(n).

Cf. A139382 (q=2), A333142.

qStirling2 := proc(n, k, q) option remember; with(QDifferenceEquations):
if n = 0 then return 0^k fi; if k = 0 then return n^0 fi;
qStirling2(n-1, k-1, p) + QBrackets(k+1, p)*qStirling2(n-1, k, p);
subs(p = q, expand(%)) end:
seq(seq(qStirling2(n, k, 3), k=0..n), n=0..9);

qStirling2[n_, k_, q_] /; 1 <= k <= n := (* q^(k-1) *) qStirling2[n - 1, k - 1, q] + Sum[q^j, {j, 0, k - 1}] qStirling2[n - 1, k, q];
qStirling2[n_, 0, _] := KroneckerDelta[n, 0];
qStirling2[0, k_, _] := KroneckerDelta[0, k];
qStirling2[, , _] = 0;
Table[qStirling2[n + 1, k + 1, 3], {n, 0, 9}, {k, 0, n}] // Flatten (* Jean-François Alcover, Mar 11 2020 *)

qStirling2(n, k, q) = qStirling2(n-1, k-1, q) + qBrackets(k+1, q)*qStirling2(n-1, k, q) with boundary values 0^k if n = 0 and n^0 if k = 0.

Note that also a second definition is used in the literature which has an additional factor q^k attached to the first term in the equation above. The two versions differ by a factor of q^binomial(k,2).

A353094 a(1) = 2; for n > 1, a(n) = 3*a(n-1) + 3 - n.

Original entry on oeis.org

2, 7, 21, 62, 184, 549, 1643, 4924, 14766, 44291, 132865, 398586, 1195748, 3587233, 10761687, 32285048, 96855130, 290565375, 871696109, 2615088310, 7845264912, 23535794717, 70607384131, 211822152372, 635466457094, 1906399371259, 5719198113753, 17157594341234

Offset: 1

Seiichi Manyama, Apr 23 2022

Index entries for linear recurrences with constant coefficients, signature (5,-7,3).

Cf. A064617, A353095, A353096, A353097, A353098, A353099, A353100.

Cf. A000340.

LinearRecurrence[{5, -7, 3}, {2, 7, 21}, 28] (* Amiram Eldar, Apr 23 2022 *)

my(N=30, x='x+O('x^N)); Vec(x*(2-3*x)/((1-x)^2*(1-3*x)))

PARI
```
a(n) = (3^(n+1)+2*n-3)/4;
```

b(n, k) = sum(j=0, n-1, (k-n+j)*k^j);
a(n) = b(n, 3);

G.f.: x * (2 - 3*x)/((1 - x)^2 * (1 - 3*x)).
a(n) = 5*a(n-1) - 7*a(n-2) + 3*a(n-3).
a(n) = A000340(n-1) + n.
a(n) = (3^(n+1) + 2*n - 3)/4.
a(n) = Sum_{k=0..n-1} (3 - n + k) * 3^k.
E.g.f.: exp(x)*(3*exp(2*x) + 2*x - 3)/4. - Stefano Spezia, May 28 2023

A008971 Triangle read by rows: T(n,k) is the number of permutations of [n] with k increasing runs of length at least 2.

Original entry on oeis.org

1, 1, 1, 1, 1, 5, 1, 18, 5, 1, 58, 61, 1, 179, 479, 61, 1, 543, 3111, 1385, 1, 1636, 18270, 19028, 1385, 1, 4916, 101166, 206276, 50521, 1, 14757, 540242, 1949762, 1073517, 50521, 1, 44281, 2819266, 16889786, 17460701, 2702765, 1, 132854, 14494859

Offset: 0

N. J. A. Sloane

Row n has 1+floor(n/2) terms.
T(n,k) is also the number of permutations of [n] with k "exterior peaks" where we count peaks in the usual way, but add a peak at the beginning if the permutation begins with a descent, e.g. 213 has one exterior peak. T(3,1) = 5 since all permutations of [3] have an exterior peak except 123. This is also the definition for peaks of signed permutations, where we assume that a signed permutation always begins with a zero. - Kyle Petersen, Jan 14 2005
From Petros Hadjicostas, Aug 09 2019: (Start)
In their book, David and Barton (1962) use the notation T_{N,v*}^* for this array, where N is the length of the permutation and v* is the so-called "number of runs up" in the permutation. In modern terminology, a "run up" in a permutation is an increasing run of length >= 2. See their discussion on p. 154 of their book and see p. 163 for the definition of T_{N,v*}^*.
They do not consider as "runs up" single elements b_i in a permutation b = (b_1, b_2, ..., b_n) even if they satisfy b_{i-1} > b_i > b_{i+1} (with b_{n-1} > b_n when i = n and b_1 > b_2 when i = 1). (The command Runs[b] for permutation b in Mathematica, using the package Combinatorica`, will generate not only the "runs up" of b but also the single elements in the permutation b that satisfy one of the above inequalities. For example, Runs[{3,2,1}] gives the set of runs {{3}, {2}, {1}}, none of which is a "run up".)
So, here n = N and k = v*. On p. 163 of their book they give the recurrence shown below in the FORMULA section from Charalambides' (2002) book: T(n+1, k) = (2*k + 1) * T(n,k) + (n - 2*k + 2) * T(n, k-1) for n >= 0 and 1 <= k <= floor(n/2) + 1. The values of T_{N,v*}^* = T(n, k) appear in Table 10.5 (p. 163) of their book.
Since the complement of a permutation (b_1, b_2, ..., b_n) is (n+1-b_1, n+1-b_2, ..., n+1-b_n), it is clear that the current array T(n, k) is also the number of permutations of [n] with k decreasing runs of length >= 2 (i.e., the number of permutations of [n] with k "runs down" according to David and Barton (1962)).
Note that the number of permutations of [n] with k runs of length >= 2 that are either increasing or decreasing (i.e., the number of permutations of [n] that contain k "runs up" and "runs down" in total) is given by array A059427. One half of array A059427 is given in Table 10.4 (p. 159) in David and Barton (1962)--see also array A008970.
A run that is either a "run up" or "run down" (i.e., an ascending or a descending run of length >= 2) is called "séquence" by André (19th century) and Comtet (1974). See the references for sequence A000708 or for array A059427. (Again, recall that David and Barton do not consider single numbers as either a "run up" or a "run down".)
Morley (1897) proved that in a permutation of [n], #("runs up") + #("runs down") + #(monotonic triplets of adjacent numbers in the permutation) = n - 1. (His definition of a run is highly non-standard!) See the example below.
The number Q(n,k) of circular permutations of [n] that contain k runs that are either "runs up" or "runs down" (that is, k ascending or descending runs of length >= 2) is given by array A008303. More precisely, Q(n+1, 2*(k+1)) = A008303(n, k) for n >= 1 and 0 <= k <= ceiling(n/2)-1. In addition, Q(n, s) = 0 when either s is odd, or n <= 1, or s > n. Also, Q_{n,2} = 2^(n-2) for n >= 2.
The numbers in array A008303 appear in Table 10.6 (p. 163) in David and Barton (1962).
(End)

			Triangle T(n,k) (with rows n >= 0 and columns k >= 0) starts as follows:
  1;
  1;
  1,   1;
  1,   5;
  1,  18,       5;
  1,  58,      61;
  1, 179,     479,     61;
  1, 543,    3111,   1385;
  1, 1636,  18270,  19028,  1385;
  1, 4916, 101166, 206276, 50521;
  ...
Example: T(3,1) = 5 because all permutations of [3] with the exception of 321 have one increasing run of length at least 2.
From _Peter Bala_, Jan 23 2016: (Start)
Row 6: cos(x)^7*(d/dx)^6(1/cos(x)) = sin(x)^6 + 179*sin(x)^4 + 479*sin(x)^2 + 61.
Equivalently, (sqrt(1 - x^2))^7*D^6(1/sqrt(1 - x^2)) = x^6 + 179*x^4 + 479*x^2 + 61, where D = sqrt(1 - x^2)*d/dx. (End)
From _Petros Hadjicostas_, Aug 09 2019: (Start)
Consider the permutations of [4]. We list the increasing runs of length at least 2 (= "runs up"), the decreasing runs of length at least 2 (= "runs down"), and the monotonic triplets of adjacent numbers in the permutation (which we abbreviate to MTAN for simplicity). The sum of the numbers of these three should be n-1 (see Morley (1897) but notice that his use of the word "run" is highly non-standard).
1234 -> "runs up" = {1234}, "runs down" = {}, MTAN = {123, 234}.
1243 -> "runs up" = {124}, "runs down" = {43}, MTAN = {124}.
1324 -> "runs up" = {13, 24}, "runs down" = {32}, MTAN = {}.
1342 -> "runs up" = {134}, "runs down" = {42}, MTAN = {134}.
1423 -> "runs up" = {14, 23}, "runs down" = {42}, MTAN = {}.
1432 -> "runs up" = {14}, "runs down" = {432}, MTAN = {432}.
2134 -> "runs up" = {134}, "runs down" = {21}, MTAN = {134}.
2143 -> "runs up" = {14}, "runs down" = {21, 43}, MTAN = {}.
2314 -> "runs up" = {23, 14}, "runs down" = {31}, MTAN = {}.
2341 -> "runs up" = {234}, "runs down" = {41}, MTAN = {234}.
2413 -> "runs up" = {24, 13}, "runs down" = {41}, MTAN = {}.
2431 -> "runs up" = {24}, "runs down" = {431}, MTAN = {431}.
3124 -> "runs up" = {124}, "runs down" = {31}, MTAN = {124}.
3142 -> "runs up" = {14}, "runs down" = {31, 42}, MTAN = {}.
3214 -> "runs up" = {14}, "runs down" = {321}, MTAN = {321}.
3241 -> "runs up" = {24}, "runs down" = {32, 41}, MTAN = {}.
3412 -> "runs up" = {34, 12}, "runs down" = {41}, MTAN = {}.
3421 -> "runs up" = {34}, "runs down" = {421}, MTAN = {421}.
4123 -> "runs up" = {123}, "runs down" = {41}, MTAN = {123}.
4132 -> "runs up" = {13}, "runs down" = {41, 32}, MTAN = {}.
4213 -> "runs up" = {13}, "runs down" = {421}, MTAN = {421}.
4231 -> "runs up" = {23}, "runs down" = {42, 31}, MTAN = {}.
4312 -> "runs up" = {12}, "runs down" = {431}, MTAN = {431}.
4321 -> "runs up" = {}, "runs down" = {4321}, MTAN = {432, 321}.
If we let k = number of increasing runs of length >= 2 (= number of "runs up") in a permutation of [4], then (from above) the possible values of k are 0, 1, 2, and we have T(n=4, k=0) = 1, T(n=4, k=1) = 18, and T(n=4, k=2) = 5.
If we let k = number of decreasing runs of length >= 2 (= number of "runs down") in a permutation of [4], then again the possible values of k are 0, 1, 2, and we have T(n=4, k=0) = 1, T(n=4, k=1) = 18, and T(n=4, k=2) = 5.
Finally, note that if b_i, b_{i+1}, b_{i+2} is an increasing triplet of adjacent numbers in permutation b, then n+1-b_i, n+1-b_{i+1}, n+1-b_{i+2} is a decreasing triplet of adjacent numbers in the complement of b, and vice versa. For example, 4213 is the complement of 1342. Their set of monotonic triplets of adjacent numbers are {421} and {134}, respectively, and we have 4 + 1 = 2 + 3 = 1 + 4 = 5.
(End)

Ch. A. Charalambides, Enumerative Combinatorics, Chapman & Hall/CRC, Boca Raton, Florida, 2002.
F. N. David and D. E. Barton, Combinatorial Chance, Charles Griffin, 1962; see Table 10.5, p. 163.
F. N. David, M. G. Kendall and D. E. Barton, Symmetric Function and Allied Tables, Cambridge, 1966, p. 260.

Alois P. Heinz, Rows n = 0..170, flattened
David Callan, Shi-Mei Ma, and Toufik Mansour, Some Combinatorial Arrays Related to the Lotka-Volterra System, Electronic Journal of Combinatorics, Volume 22, Issue 2 (2015), Paper #P2.22.
Shaoshi Chen, Hanqian Fang, Sergey Kitaev, and Candice X.T. Zhang, Patterns in Multi-dimensional Permutations, arXiv:2411.02897 [math.CO], 2024. See p. 7.
C.-O. Chow and S.-M. Ma, Counting signed permutations by their alternating runs, Discrete Mathematics, Volume 323, 28 May 2014, Pages 49-57.
C.-O. Chow, S.-M. Ma, T. Mansour, and M. Shattuck, Counting permutations by cyclic peaks and valleys, Annales Mathematicae et Informaticae, (2014), Vol. 43, pp. 43-54.
Ming-Jian Ding and Bao-Xuan Zhu, Some results related to Hurwitz stability of combinatorial polynomials, Advances in Applied Mathematics, Volume 152, (2024), 102591. See p. 13.
Qi Fang, Ya-Nan Feng, and Shi-Mei Ma, Alternating runs of permutations and the central factorial numbers, arXiv:2202.13978 [math.CO], 2022.
FindStat - Combinatorial Statistic Finder, The number of outer peaks of a permutation
Amy M. Fu, A Context-free Grammar for Peaks and Double Descents of Permutations, arXiv:1801.04397 [math.CO], 2018.
Amy M. Fu and Frank Z. K. Li, Joint Distributions of Permutation Statistics and the Parabolic Cylinder Functions, arXiv:1809.07465 [math.CO], 2018.
D. S. Hollman and H. F. Schaefer III, Arbitrary order El'yashevich-Wilson B tensor formulas for the most frequently used internal coordinates in molecular vibrational analyses, The Journal of Chemical Physics, Vol. 137, 164103 (2012). - From _N. J. A. Sloane_, Jan 01 2013
Kathy Q. Ji, The (alpha, beta)-Eulerian Polynomials and Descent-Stirling Statistics on Permutations, arXiv:2310.01053 [math.CO], 2023. See pp. 9, 22.
Shi-Mei Ma, Derivative polynomials and permutations by numbers of interior peaks and left peaks, arXiv preprint arXiv:1106.5781 [math.CO], 2011.
Shi-Mei Ma, Enumeration of permutations by number of alternating runs, Discrete Math., 313 (2013), 1816-1822.
Shi-Mei Ma, Qi Fang, Toufik Mansour, and Yeong-Nan Yeh, Alternating Eulerian polynomials and left peak polynomials, arXiv:2104.09374 [math.CO], 2021.
Shi-Mei Ma, T. Mansour, and D. Callan, Some combinatorial arrays related to the Lotka-Volterra system, arXiv preprint arXiv:1404.0731 [math.CO], 2014.
S.-M. Ma, T. Mansour, and D. G. L. Wang, Combinatorics of Dumont differential system on the Jacobi elliptic functions, arXiv preprint arXiv:1403.0233 [math.CO], 2014.
Shi-Mei Ma, Toufik Mansour, David G.L. Wang, and Yeong-Nan Yeh, Several variants of the Dumont differential system and permutation statistics, Science China Mathematics 60 (2018).
Shi-Mei Ma, Jun Ma, Jean Yeh, and Yeong-Nan Yeh, The 1/k-Eulerian polynomials of type B, arXiv:2001.07833 [math.CO], 2020.
Shi-Mei Ma and Yeong-Nan Yeh, The Peak Statistics on Simsun Permutations, Elect. J. Combin., 23 (2016), P2.14; arXiv preprint, arXiv:1601.06505 [math.CO], 2016.
F. Morley, A generating function for the number of permutations with an assigned number of sequences, Bull. Amer. Math. Soc. 4 (1897), 23-28.
L. W. Shapiro, W.-J. Woan, and S. Getu, Runs, slides and moments, SIAM J. Alg. Discrete Methods, 4 (1983), 459-466.
Wikipedia, Florence Nightingale David.
Bao-Xuan Zhu, Stieltjes moment properties and continued fractions from combinatorial triangles, arXiv:2007.14924 [math.CO], 2020, see p. 27.
Yan Zhuang, Monoid networks and counting permutations by runs, arXiv preprint arXiv:1505.02308 [math.CO], 2015.
Y. Zhuang, Counting permutations by runs, J. Comb. Theory Ser. A 142 (2016), pp. 147-176.

A160486 is a sub-triangle.
A000340, A000363, A000507 equal the second, third and fourth left hand columns.
Cf. A000708, A008303, A008970, A059427.
T(2n,n) gives A000364.

G:=sqrt(1-t)/(sqrt(1-t)*cosh(x*sqrt(1-t))-sinh(x*sqrt(1-t))): Gser:=simplify(series(G,x=0,15)): for n from 0 to 14 do P[n]:=sort(expand(n!*coeff(Gser,x,n))) od: seq(seq(coeff(t*P[n],t^k),k=1..1+floor(n/2)),n=0..14); # edited by Johannes W. Meijer, May 15 2009
# second Maple program:
T:= proc(n, k) option remember; `if`(k=0, 1, `if`(k>iquo(n, 2), 0,
      (2*k+1)*T(n-1, k) +(n+1-2*k)*T(n-1, k-1)))
    end:
seq(seq(T(n, k), k=0..n/2), n=0..14); # Alois P. Heinz, Oct 16 2013

t[n_, 0] = 1; t[n_, k_] /; k > Floor[n/2] = 0;
t[n_ , k_ ] /; k <= Floor[n/2] := t[n, k] = (2 k + 1) t[n - 1, k] + (n - 2 k + 1) t[n - 1, k - 1];
Flatten[Table[t[n, k], {n, 0, 12}, {k, 0, Floor[n/2]}]][[1 ;; 45]] (* Jean-François Alcover, May 30 2011, after given formula *)

E.g.f.: G(t,x) = sum(T(n,k) t^k x^n/n!, 0<=k<=floor(n/2), n>=0) = sqrt(1-t)/(sqrt(1-t)*cosh(x*sqrt(1-t))-sinh(x*sqrt(1-t))) (Ira M. Gessel).
T(n+1,k) = (2*k+1)*T(n,k) + (n-2*k+2)*T(n,k-1) (see p. 542 of the Charalambides reference or p. 163 in the David and Barton book).
G.f.: T(0)/(1-x), where T(k) = 1 - y*x^2*(k+1)^2/(y*x^2*(k+1)^2 - (1 -x -2*x*k)*(1 -3*x -2*x*k)/T(k+1) ); (continued fraction). - Sergei N. Gladkovskii, Nov 08 2013
From Peter Bala, Jan 23 2016: (Start)
cos(x)^(n+1)*(d/dx)^n(1/cos(x)) = Sum_{k = 0..floor(n/2)} T(n,k)*sin(x)^(n-2*k).
Equivalently, let D be the differential operator sqrt(1 - x^2)*d/dx. Then sqrt(1 - x^2)^(n+1)*D^n(1/sqrt(1 - x^2)) = Sum_{k = 0..floor(n/2)} T(n,k)*x^(n-2*k). (End)

Edited by Emeric Deutsch and Ira M. Gessel, Aug 28 2004

Crossrefs added by Johannes W. Meijer, May 24 2009

A061981 a(n) = 3^n - 2*n - 1.

Original entry on oeis.org

0, 0, 4, 20, 72, 232, 716, 2172, 6544, 19664, 59028, 177124, 531416, 1594296, 4782940, 14348876, 43046688, 129140128, 387420452, 1162261428, 3486784360, 10460353160, 31381059564, 94143178780, 282429536432, 847288609392

Offset: 0

Henry Bottomley, May 24 2001

Harry J. Smith, Table of n, a(n) for n = 0..200
Brandon Alberts and Jack Klys, The distribution of H8-extensions of quadratic fields, arXiv:1611.05595 [math.NT], 2016. See p. 17.
Index entries for linear recurrences with constant coefficients, signature (5,-7,3).

Column of A061980.

Cf. A000340, A186948.

Table[3^n-2n-1,{n,0,30}] (* or *) LinearRecurrence[{5,-7,3},{0,0,4},30] (* Harvey P. Dale, Mar 30 2018 *)

a(n) = { 3^n - 2*n - 1 } \\ Harry J. Smith, Jul 29 2009

[3^n -(2*n+1) for n in (0..40)] # G. C. Greubel, Jun 13 2022

From Bruno Berselli, Jan 31 2012: (Start)
G.f.: 4*x^2/((1-3*x)*(1-x)^2).
a(n) = A186948(n) - 1.
a(n+2) = 4*A000340(n). (End)
From Deisy J. Camacho, Feb 26 2021 (Start)
a(n) = Sum_{j=2..n} Sum_{i=0..j} n!/((j-i)!*i!*(n-j)!).
a(n) = 4 + 4*a(n-1) - 3*a(n-2). (End)
E.g.f.: exp(3*x) - (2*x + 1)*exp(x). - G. C. Greubel, Jun 13 2022

A102365 Triangle T(n,k), 0 <= k <= n, read by rows: given by [ 1, 0, 3, 0, 5, 0, 7, 0, 9, 0, ...] DELTA [ 0, 1, 0, 2, 0, 3, 0, 4, 0, 5, 0, ...] where DELTA is the operator defined in A084938.

Original entry on oeis.org

1, 1, 0, 1, 1, 0, 1, 5, 1, 0, 1, 18, 15, 1, 0, 1, 58, 129, 37, 1, 0, 1, 179, 877, 646, 83, 1, 0, 1, 543, 5280, 8030, 2685, 177, 1, 0, 1, 1636, 29658, 82610, 56285, 10002, 367, 1, 0, 1, 4916, 159742, 756218, 919615, 335162, 34777, 749, 1, 0

Offset: 0

Philippe Deléham, Feb 22 2005

Generalized Eulerian numbers A008292.

Reversal of A211399. - Philippe Deléham, Feb 12 2013

			Triangle begins:
  1;
  1,  0;
  1,  1,   0;
  1,  5,   1,  0;
  1, 18,  15,  1, 0;
  1, 58, 129, 37, 1, 0; ...

G. C. Greubel, Table of n, a(n) for the first 50 rows, flattened

Diagonals: A000007, A000012, A050488, A142965, A142966.

Columns: A000012, A000340, A156922, A156923, A156924.

T[0, 0] := 1;  T[n_, -1] := 0;  T[n_, n_] := 0; T[n_, k_] := T[n, k] = (n - k)*T[n - 1, k - 1] + (2*k + 1)*T[n - 1, k]; Join[{1}, Table[If[k < 0, 0, If[k >= n, 0, T[n, k]]], {n, 1, 5}, {k, 0, n}] // Flatten] (* G. C. Greubel, Jun 30 2017 *)

T(n, k) = (n-k)*T(n-1, k-1) + (2*k+1)*T(n-1, k) with T(0, 0) = 1, T(0, k) = 0 if k > 0, T(n, k) = 0 if k < 0.
Sum_{k>=0} T(n, k)*2^k = A001147(n).
Sum_{k>=0} T(n, k) = A014307(n). - Philippe Deléham, Mar 19 2005

A160486 Triangle of polynomial coefficients related to the o.g.f.s. of the RBS1 polynomials.

Original entry on oeis.org

1, 1, 1, 1, 18, 5, 1, 179, 479, 61, 1, 1636, 18270, 19028, 1385, 1, 14757, 540242, 1949762, 1073517, 50521, 1, 132854, 14494859, 137963364, 241595239, 82112518, 2702765

Offset: 1

Johannes W. Meijer, May 24 2009, Sep 19 2012

As we showed in A160485 the n-th term of the coefficients of matrix row BS1[1-2*m,n] for m = 1 , 2, 3, .. , can be generated with the RBS1(1-2*m,n) polynomials.
We define the o.g.f.s. of these polynomials by GFRBS1(z,1-2*m) = sum(RBS1(1-2*m,n)*z^(n-1), n=1..infinity) for m = 1, 2, 3, .. . The general expression of these o.g.f.s. is GFRBS1(z,1-2*m) = (-1)*RB(z,1-2*m)/(z-1)^m.
The RB(z,1-2*m) polynomials lead to a triangle that is a subtriangle of the 'double triangle' A008971. The even rows of the latter triangle are identical to the rows of our triangle.
The Maple program given below is derived from the one given in A008971.

			The first few rows of the triangle are:
[1]
[1, 1]
[1, 18, 5]
[1, 179, 479, 61]
[1, 1636, 18270, 19028, 1385]
The first few RB(z,1-2*m) polynomials are:
RB(z,-1) = 1
RB(z,-3) = z+1
RB(z,-5) = z^2+18*z+5
RB(z,-7) = z^3+179*z^2+479*z+61
The first few GFRBS1(z,1-2*m) are:
GFRBS1(z,-1) = (-1)*(1)/(z-1)
GFRBS1(z,-3) = (-1)*(z+1)/(z-1)^2
GFRBS1(z,-5) = (-1)*(z^2+18*z+5)/(z-1)^3
GFRBS1(z,-7) = (-1)*(z^3+179*z^2+479*z+61)/(z-1)^4

Cf. A160480 and A160485.
The row sums equal A010050.
This triangle is a sub-triangle of A008971.
A000340(2*n-2), A000363(2*n+2) and A000507(2*n+4) equal the second, third and fourth left hand columns.
The first right hand column equals the Euler numbers A000364.

nmax:=15; G := sqrt(1-t)/(sqrt(1-t)*cosh(x*sqrt(1-t))-sinh(x*sqrt(1-t))): Gser := simplify(series(G, x=0, nmax+1)): for m from 0 to nmax do P[m] := sort(expand(m!* coeff(Gser, x, m))) od: nmx := floor(nmax/2); for n from 0 to nmx do for k from 0 to nmx-1 do A(n+1, n+1-k) := coeff(P[2*n], t, n-k) od: od: seq(seq(A(n,m), m=1..n), n=1..nmx);

A211399 Triangle T(n,k), 0 <= k <= n, given by (0, 1, 0, 2, 0, 3, 0, 4, 0, 5, 0, ...) DELTA (1, 0, 3, 0, 5, 0, 7, 0, 9, 0, ...) where DELTA is the operator defined in A084938.

Original entry on oeis.org

1, 0, 1, 0, 1, 1, 0, 1, 5, 1, 0, 1, 15, 18, 1, 0, 1, 37, 129, 58, 1, 0, 1, 83, 646, 877, 179, 1, 0, 1, 177, 2685, 8030, 5280, 543, 1, 0, 1, 367, 10002, 56285, 82610, 29658, 1636, 1, 0, 1, 749, 34777, 335162

Offset: 0

Philippe Deléham, Feb 08 2013

Contains A156920 as submatrix.

Row-reversal of A102365. - Philippe Deléham, Feb 12 2013

			Triangle begins :
1
0, 1
0, 1, 1
0, 1, 5, 1
0, 1, 15, 18, 1
0, 1, 37, 129, 58, 1
0, 1, 83, 646, 877, 179, 1

Left hand column sequences: A000007, A000012, A050488, A142965, A142966, A142968.
Right hand column sequences: A000340, A156922, A156923, A156924.
Row sums A014307(n).

T(n,k) = k*T(n-1,k) + (2n-2k+1)*T(n-1,k-1) , T(n,n) = 1, T(n,k) = 0 if k<0 or if k>n.
T(n,k) = A185411(n,k)/(2^(n-k)).
Sum_{k, 0<=k<=n} T(n,k)*x^(n-k) = A000012(n), A014307(n), A001147(n) for x = 0, 1, 2 respectively .
G.f.: 1/(1-xy/(1-x/(1-3xy/(1-2x/(1-5xy/(1-3x/(1-7xy/(1- ...(continued fraction).

A262592 a(n) = (3^(n+1) - 2n^2 + 4n + 5) / 8.

Original entry on oeis.org

1, 2, 4, 10, 29, 88, 268, 812, 2449, 7366, 22124, 66406, 199261, 597836, 1793572, 5380792, 16142465, 48427498, 145282612, 435847970, 1307544061, 3922632352, 11767897244, 35303691940, 105911076049, 317733228398, 953199685468, 2859599056702, 8578797170429, 25736391511636

Offset: 0

N. J. A. Sloane, Oct 21 2015

Colin Barker, Table of n, a(n) for n = 0..1000
K. Satyanarayana, Sequences whose kth differences form a geometrical progression, Math. Student, 12 (1944), page 109. [Annotated scanned copy. This sequence was formerly A2752 but has now been renumbered]
Index entries for linear recurrences with constant coefficients, signature (6,-12,10,-3).

Other sequences with generating functions like this: A000340, A052161, A262593, A262594.

f1:=(a,b)->(1-a*x)^a/((1-x)^b*(1-b*x));
f2:=(a,b)->seriestolist(series(f1(a,b),x,40));
f2(2,3);

Table[3^(n + 1)/8 + 5/8 - n^2/4 + n/2, {n, 0, 29}] (* Michael De Vlieger, Oct 23 2015 *)
LinearRecurrence[{6,-12,10,-3},{1,2,4,10},30] (* Harvey P. Dale, Jul 18 2025 *)

a(n) = 3^(n+1)/8+5/8-n^2/4+n/2 \\ Colin Barker, Oct 23 2015

Vec((1-2*x)^2/((1-x)^3*(1-3*x)) + O(x^40)) \\ Colin Barker, Oct 23 2015

G.f.: (1-2*x)^2/((1-x)^3*(1-3*x)).

a(n) = 6*a(n-1)-12*a(n-2)+10*a(n-3)-3*a(n-4) for n>3. - Colin Barker, Oct 23 2015

cp's OEIS Frontend

A156923 Fourth right hand column (n-m=3) of the A156920 triangle.

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A156924 Fifth right hand column (n-m=4) of the A156920 triangle.

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A333143 Triangle read by rows: T(n, k) = qStirling2(n, k, q) for q = 3, with 0 <= k <= n.

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A353094 a(1) = 2; for n > 1, a(n) = 3*a(n-1) + 3 - n.

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A008971 Triangle read by rows: T(n,k) is the number of permutations of [n] with k increasing runs of length at least 2.

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A061981 a(n) = 3^n - 2*n - 1.

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A102365 Triangle T(n,k), 0 <= k <= n, read by rows: given by [ 1, 0, 3, 0, 5, 0, 7, 0, 9, 0, ...] DELTA [ 0, 1, 0, 2, 0, 3, 0, 4, 0, 5, 0, ...] where DELTA is the operator defined in A084938.

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A160486 Triangle of polynomial coefficients related to the o.g.f.s. of the RBS1 polynomials.

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Maple