A101494 -id:A101494 - cp's OEIS frontend

A101495 Column 1 of triangle A101494.

1, 1, 3, 8, 23, 73, 253, 948, 3817, 16433, 75295, 365600, 1874083, 10108025, 57194585, 338615084, 2092609701, 13470059649, 90137761867, 625940219896, 4503468629391, 33520255302185, 257764170849941, 2045255722438180

Offset: 0

Paul D. Hanna, Jan 21 2005

nonn

a(n) is the number of m-tuples of nonnegative integers in which n-m+1 is the maximum and appears exactly once. - Mathew Englander, Apr 11 2021

Muniru A Asiru, Table of n, a(n) for n = 0..590
Mathew Englander, Comments on A101494 and A089246, and related sequences

Cf. A101494.

List([0..30],n->Sum([0..n],k->(k+1)*(n-k)^k)); # Muniru A Asiru, Mar 05 2019

[(&+[(k+1)*(n-k)^k: k in [0..n]]): n in [0..30]]; // G. C. Greubel, Mar 05 2019

Join[{1},Table[Sum[(k+1)(n-k)^k,{k,0,n}],{n,30}]] (* Harvey P. Dale, Mar 04 2013 *)

a(n) = sum(k=0, n, (k+1)*(n-k)^k); \\ Michel Marcus, Mar 05 2019

[sum((k+1)*(n-k)^k for k in (0..n)) for n in (0..30)] # G. C. Greubel, Mar 05 2019

From Vladeta Jovovic, Sep 07 2006: (Start)
a(n) = Sum_{k=0..n} (k+1)*(n-k)^k.
G.f.: Sum_{k>=0} x^k/(1-k*x)^2. (End)

A026898 a(n) = Sum_{k=0..n} (n-k+1)^k.

Original entry on oeis.org

1, 2, 4, 9, 23, 66, 210, 733, 2781, 11378, 49864, 232769, 1151915, 6018786, 33087206, 190780213, 1150653921, 7241710930, 47454745804, 323154696185, 2282779990495, 16700904488706, 126356632390298, 987303454928973, 7957133905608837, 66071772829247410

Offset: 0

N. J. A. Sloane

nonn

Row sums of A004248, A009998, A009999.
First differences are in A047970.
First differences of A103439.
Antidiagonal sums of array A003992.
a(n-1), for n>=1, is the number of length-n restricted growth strings (RGS) [s(0),s(1),...,s(n-1)] where s(0)=0 and s(k)<=1+max(prefix) for k>=1, that are simultaneously projections as maps f: [n] -> [n] where f(x)<=x and f(f(x))=f(x); see example and the two comments (Arndt, Apr 30 2011 Jan 04 2013) in A000110. - Joerg Arndt, Mar 07 2015
Number of finite sequences s of length n+1 whose discriminator sequence is s itself. Here the discriminator sequence of s is the one where the n-th term (n>=1) is the least positive integer k such that the first n terms are pairwise incongruent, modulo k. - Jeffrey Shallit, May 17 2016
From Gus Wiseman, Jan 08 2019: (Start)
Also the number of set partitions of {1,...,n+1} whose minima form an initial interval of positive integers. For example, the a(3) = 9 set partitions are:
  {{1},{2},{3},{4}}
  {{1},{2},{3,4}}
  {{1},{2,4},{3}}
  {{1,4},{2},{3}}
  {{1},{2,3,4}}
  {{1,3},{2,4}}
  {{1,4},{2,3}}
  {{1,3,4},{2}}
  {{1,2,3,4}}
Missing from this list are:
  {{1},{2,3},{4}}
  {{1,2},{3},{4}}
  {{1,3},{2},{4}}
  {{1,2},{3,4}}
  {{1,2,3},{4}}
  {{1,2,4},{3}}
(End)
a(n) is the number of m-tuples of nonnegative integers less than or equal to n-m (including the "0-tuple"). - Mathew Englander, Apr 11 2021

			G.f.: A(x) = 1 + 2*x + 4*x^2 + 9*x^3 + 23*x^4 + 66*x^5 + 210*x^6 + ...
where we have the identity:
A(x) = 1/(1-x) + x/(1-2*x) + x^2/(1-3*x) + x^3/(1-4*x) + x^4/(1-5*x) + ...
is equal to
A(x) = 1/(1-x) + x/((1-x)^2*(1+x)) + 2!*x^2/((1-x)^3*(1+x)*(1+2*x)) + 3!*x^3/((1-x)^4*(1+x)*(1+2*x)*(1+3*x)) + 4!*x^4/((1-x)^5*(1+x)*(1+2*x)*(1+3*x)*(1+4*x)) + ...
From _Joerg Arndt_, Mar 07 2015: (Start)
The a(5-1) = 23 RGS described in the comment are (dots denote zeros):
01:  [ . . . . . ]
02:  [ . 1 . . . ]
03:  [ . 1 . . 1 ]
04:  [ . 1 . 1 . ]
05:  [ . 1 . 1 1 ]
06:  [ . 1 1 . . ]
07:  [ . 1 1 . 1 ]
08:  [ . 1 1 1 . ]
09:  [ . 1 1 1 1 ]
10:  [ . 1 2 . . ]
11:  [ . 1 2 . 1 ]
12:  [ . 1 2 . 2 ]
13:  [ . 1 2 1 . ]
14:  [ . 1 2 1 1 ]
15:  [ . 1 2 1 2 ]
16:  [ . 1 2 2 . ]
17:  [ . 1 2 2 1 ]
18:  [ . 1 2 2 2 ]
19:  [ . 1 2 3 . ]
20:  [ . 1 2 3 1 ]
21:  [ . 1 2 3 2 ]
22:  [ . 1 2 3 3 ]
23:  [ . 1 2 3 4 ]
(End)

Reinhard Zumkeller, Table of n, a(n) for n = 0..500
Fufa Beyene, Jörgen Backelin, Roberto Mantaci, and Samuel A. Fufa, Set Partitions and Other Bell Number Enumerated Objects, J. Int. Seq., Vol. 26 (2023), Article 23.1.8.
Giulio Cerbai, Pattern-avoiding modified ascent sequences, arXiv:2401.10027 [math.CO], 2024. See p. 12.
Giulio Cerbai and Anders Claesson, Counting fixed-point-free Cayley permutations, arXiv:2507.09304 [math.CO], 2025. See p. 25.
Sajed Haque, Discriminators of Integer Sequences, 2017, See p. 33 Corollary 29.
Mathematics Stack Exchange, Asymptotics of ..., 2011.
Chunyan Yan and Zhicong Lin, Inversion sequences avoiding pairs of patterns, arXiv:1912.03674 [math.CO], 2019. See p. 3.
Robin D. P. Zhou, Pattern avoidance in revised ascent sequences, arXiv:2505.05171 [math.CO], 2025. See p. 4.

Cf. A000110, A000258, A000670, A003101, A008277, A038125, A062810.

Cf. A105795, A179928, A287215, A287216.

a026898 n = sum $ zipWith (^) [n + 1, n .. 1] [0 ..]
-- Reinhard Zumkeller, Sep 14 2014

[(&+[(n-k+1)^k: k in [0..n]]): n in [0..50]]; // Stefano Spezia, Jan 09 2019

a:= n-> add((n+1-j)^j, j=0..n): seq(a(n), n=0..23); # Zerinvary Lajos, Apr 18 2009

Table[Sum[(n-k+1)^k, {k,0,n}], {n, 0, 25}] (* Michael De Vlieger, Apr 01 2015 *)

{a(n)=polcoeff(sum(m=0,n,x^m/(1-(m+1)*x+x*O(x^n))),n)} /* Paul D. Hanna, Sep 13 2011 */

{INTEGRATE(n,F)=local(G=F);for(i=1,n,G=intformal(G));G}
{a(n)=local(A=1+x);A=sum(k=0,n,INTEGRATE(k,exp((k+1)*x+x*O(x^n))));n!*polcoeff(A,n)} \\ Paul D. Hanna, Dec 28 2013
for(n=0,30,print1(a(n),", "))

{a(n)=polcoeff( sum(m=0, n, m!*x^m/(1-x +x*O(x^n))^(m+1)/prod(k=1, m, 1+k*x +x*O(x^n))), n)}  /* From o.g.f. (Paul D. Hanna, Jul 20 2014) */
for(n=0, 25, print1(a(n), ", "))

[sum((n-j+1)^j for j in (0..n)) for n in (0..30)] # G. C. Greubel, Jun 15 2021

a(n) = A003101(n) + 1.
G.f.: Sum_{n>=0} x^n/(1 - (n+1)*x). - Paul D. Hanna, Sep 13 2011
G.f.: G(0) where G(k) = 1 + x*(2*k*x-1)/((2*k*x+x-1) - x*(2*k*x+x-1)^2/(x*(2*k*x+x-1) + (2*k*x+2*x-1)/G(k+1) )); (continued fraction). - Sergei N. Gladkovskii, Jan 26 2013
E.g.f.: Sum_{n>=0} Integral^n exp((n+1)*x) dx^n, where Integral^n F(x) dx^n is the n-th integration of F(x) with no constant of integration. - Paul D. Hanna, Dec 28 2013
O.g.f.: Sum_{n>=0} n! * x^n/(1-x)^(n+1) / Product_{k=1..n} (1 + k*x). - Paul D. Hanna, Jul 20 2014
a(n) = A101494(n+1,0). - Vladimir Kruchinin, Apr 01 2015
a(n-1) = Sum_{k = 1..n} k^(n-k). - Gus Wiseman, Jan 08 2019
log(a(n)) ~ (1 - 1/LambertW(exp(1)*n)) * n * log(1 + n/LambertW(exp(1)*n)). - Vaclav Kotesovec, Jun 15 2021
a(n) ~ sqrt(2*Pi/(n+1 + (n+1)/w(n))) * ((n+1)/w(n))^(n+2 - (n+1)/w(n)), where w(n) = LambertW(exp(1)*(n+1)). - Vaclav Kotesovec, Jun 25 2021, after user "leonbloy", see Mathematics Stack Exchange link.

a(23)-a(25) from Paul D. Hanna, Dec 28 2013

A047970 Antidiagonal sums of nexus numbers (A047969).

Original entry on oeis.org

1, 2, 5, 14, 43, 144, 523, 2048, 8597, 38486, 182905, 919146, 4866871, 27068420, 157693007, 959873708, 6091057009, 40213034874, 275699950381, 1959625294310, 14418124498211, 109655727901592, 860946822538675, 6969830450679864, 58114638923638573

Offset: 0

Alford Arnold, Dec 11 1999

nonn

From Lara Pudwell, Oct 23 2008: (Start)
A permutation p avoids a pattern q if it has no subsequence that is order-isomorphic to q. For example, p avoids the pattern 132 if it has no subsequence abc with a < c < b.
Barred pattern avoidance considers permutations that avoid a pattern except in a special case. Given a barred pattern q, we may form two patterns, q1 = the sequence of unbarred letters of q and q2 = the sequence of all letters of q.
A permutation p avoids barred pattern q if every instance of q1 in p is embedded in a copy of q2 in p. In other words, p avoids q1, except in the special case that a copy of q1 is a subsequence of a copy of q2.
For example, if q=5{bar 1}32{bar 4}, then q1=532 and q2 = 51324. p avoids q if every for decreasing subsequence acd of length 3 in p, one can find letters b and e so that the subsequence abcde of p has b < d < c < e < a. (End)
Number of ordered factorizations over the Gaussian polynomials.
Apparently, also the number of permutations in S_n avoiding {bar 3}{bar 1}542 (i.e., every occurrence of 542 is contained in an occurrence of a 31542). - Lara Pudwell, Apr 25 2008
With offset 1, apparently the number of sequences {b(m)} of length n of positive integers with b(1) = 1 and, for all m > 1, b(m) <= max{b(m-1) + 1, max{b(i) | 1 <= i <= m - 1}}. This sequence begins 1, 2, 5, 14, 43, 144, 523, 2048, 8597, 38486. The term 144 counts the length 6 sequence 1, 2, 3, 1, 1, 3, for instance. Contrast with the families of sequences discussed in Franklin T. Adams-Watters's comment in A005425. - Rick L. Shepherd, Jan 01 2015
a(n-1) for n >= 1 is the number of length-n restricted growth strings (RGS) [s(0), s(1), ..., s(n-1)] with s(0)=0 and s(k) <= the number of fixed points in the prefix, see example. - Joerg Arndt, Mar 08 2015
Number of sequences (e(1), ..., e(n+1)), 0 <= e(i) < i, such that there is no triple i < j < k with e(i) != e(j) = e(k). [Martinez and Savage, 2.15] - Eric M. Schmidt, Jul 17 2017
a(n) counts all positive-integer m-tuples whose maximum is n-m+2. - Mathew Englander, Feb 28 2021
a(n) counts the cyclic permutations of [n+2] that avoid the vincular pattern 12-3-4, i.e., the pattern 1234 where the 1 and 2 are required to be adjacent. - Rupert Li, Jul 27 2021

			a(3) = 1 + 5 + 7 + 1 = 14.
From _Paul D. Hanna_, Jul 22 2014:  (Start)
G.f. A(x) = 1 + 2*x + 5*x^2 + 14*x^3 + 43*x^4 + 144*x^5 + 523*x^6 + 2048*x^7 + ...
where we have the series identity:
A(x) = (1-x)*( 1/(1-2*x) + x/(1-3*x) + x^2/(1-4*x) + x^3/(1-5*x) + x^4/(1-6*x) + x^5/(1-7*x) + x^6/(1-8*x) + ...)
is equal to
A(x) = 1/(1-x) + x/((1-x)*(1-2*x)) + x^2/((1-2*x)*(1-3*x)) + x^3/((1-3*x)*(1-4*x)) + x^4/((1-4*x)*(1-5*x)) + x^5/((1-5*x)*(1-6*x)) + x^6/((1-6*x)*(1-7*x)) + ...
and also equals
A(x) = 1/((1-x)*(1+x)) + 2!*x/((1-x)^2*(1+x)*(1+2*x)) + 3!*x^2/((1-x)^3*(1+x)*(1+2*x)*(1+3*x)) + 4!*x^3/((1-x)^4*(1+x)*(1+2*x)*(1+3*x)*(1+4*x)) + ...
(End)
From _Joerg Arndt_, Mar 08 2015: (Start)
There are a(4-1)=14 length-4 RGS as in the comment (dots denote zeros):
01:  [ . . . . ]
02:  [ . . . 1 ]
03:  [ . . 1 . ]
04:  [ . . 1 1 ]
05:  [ . 1 . . ]
06:  [ . 1 . 1 ]
07:  [ . 1 . 2 ]
08:  [ . 1 1 . ]
09:  [ . 1 1 1 ]
10:  [ . 1 1 2 ]
11:  [ . 1 2 . ]
12:  [ . 1 2 1 ]
13:  [ . 1 2 2 ]
14:  [ . 1 2 3 ]
(End)

Alois P. Heinz, Table of n, a(n) for n = 0..300
G. E. Andrews, The Theory of Partitions, 1976, page 242 table of Gaussian polynomials.
David Callan, The number of bar(31)542-avoiding permutations, arXiv:1111.3088 [math.CO], 2011.
Rupert Li, Vincular Pattern Avoidance on Cyclic Permutations, arXiv:2107.12353 [math.CO], 2021.
Zhicong Lin and Sherry H. F. Yan, Vincular patterns in inversion sequences, Applied Mathematics and Computation (2020), Vol. 364, 124672.
Megan A. Martinez and Carla D. Savage, Patterns in Inversion Sequences II: Inversion Sequences Avoiding Triples of Relations, arXiv:1609.08106 [math.CO], 2016.
Lara Pudwell, Enumeration Schemes for Pattern-Avoiding Words and Permutations, Ph. D. Dissertation, Math. Dept., Rutgers University, May 2008.
Lara Pudwell, Enumeration schemes for permutations avoiding barred patterns, El. J. Combinat. 17 (1) (2010) R29.
Eric Weisstein's World of Mathematics, Nexus Number
Chunyan Yan and Zhicong Lin, Inversion sequences avoiding pairs of patterns, arXiv:1912.03674 [math.CO], 2019.

Antidiagonal sums of A085388 (beginning with the second antidiagonal) and A047969.
Partial sums are in A026898, A003101. First differences A112532.
Cf. A112531, A089246, A242431, A101494.

T := proc(n, k) option remember; local j;
    if k=n then 1
  elif k>n then 0
  else (k+1)*T(n-1, k) + add(T(n-1, j), j=k..n)
    fi end:
A047970 := n -> T(n,0);
seq(A047970(n), n=0..24); # Peter Luschny, May 14 2014

a[ n_] := SeriesCoefficient[ ((1 - x) Sum[ x^k / (1 - (k + 2) x), {k, 0, n}]), {x, 0, n}]; (* Michael Somos, Jul 09 2014 *)

/* From o.g.f. (Paul D. Hanna, Jul 20 2014) */
{a(n)=polcoeff( sum(m=0, n, (m+1)!*x^m/(1-x)^(m+1)/prod(k=1, m+1, 1+k*x +x*O(x^n))), n)}
for(n=0, 25, print1(a(n), ", "))

/* From o.g.f. (Paul D. Hanna, Jul 22 2014) */
{a(n)=polcoeff( sum(m=0, n, x^m/((1-m*x)*(1-(m+1)*x +x*O(x^n)))), n)}
for(n=0, 25, print1(a(n), ", "))

def A074664():
    T = []; n = 0
    while True:
        T.append(1)
        yield T[0]
        for k in (0..n):
            T[k] = (k+1)*T[k] + add(T[j] for j in (k..n))
        n += 1
a = A074664()
[next(a) for n in range(25)] # Peter Luschny, May 13 2014

Formal o.g.f.: (1 - x)*( Sum_{n >= 0} x^n/(1 - (n + 2)*x) ). - Peter Bala, Jul 09 2014
O.g.f.: Sum_{n>=0} (n+1)! * x^n/(1-x)^(n+1) / Product_{k=1..n+1} (1 + k*x). - Paul D. Hanna, Jul 20 2014
O.g.f.: Sum_{n>=0} x^n / ( (1 - n*x) * (1 - (n+1)*x) ). - Paul D. Hanna, Jul 22 2014
From Mathew Englander, Feb 28 2021: (Start)
a(n) = A089246(n+2,0) = A242431(n,0).
a(n) = Sum_{m = 1..n+1} Sum_{i = 0..m-1} binomial(m,i) * (n-m+1)^i.
a(n) = 1 + Sum_{i = 0..n} i * (i+1)^(n-i). (End)
a(n) ~ sqrt(2*Pi*n / (w*(1+w))) * (1 + n/w)^(1 + n - n/w), where w = LambertW(exp(1)*n). - Vaclav Kotesovec, Jun 10 2025

A089246 Triangular array read by rows: a(n, k) is the number of ordered m-tuples of positive integers (x_1, ..., x_m) such that max x_i = n+1-m and there are k ones (0 <= k <= n).

Original entry on oeis.org

1, 0, 1, 1, 0, 1, 2, 2, 0, 1, 5, 5, 3, 0, 1, 14, 15, 9, 4, 0, 1, 43, 50, 31, 14, 5, 0, 1, 144, 180, 118, 54, 20, 6, 0, 1, 523, 695, 481, 229, 85, 27, 7, 0, 1, 2048, 2869, 2081, 1035, 395, 125, 35, 8, 0, 1, 8597, 12616, 9535, 4929, 1951, 629, 175, 44, 9, 0, 1, 38486, 58862

Offset: 0

Alford Arnold, Dec 22 2003

The row sums are given by A047970 because row n counts the same tuples as the n-th antidiagonal of A047969.

			a(5, 0) = 14: (5), (4,2), (2,4), (4,3), (3,4), (4,4),
(3,2,2), (2,3,2), (2,2,3), (3,3,2), (3,2,3), (2,3,3), (3,3,3), (2,2,2,2).
a(5, 1) = 15: (4,1), (1,4), (3,3,1), (3,1,3), (1,3,3), 6 permutations of (3,2,1) and 4 permutations of (2,2,2,1).
Triangle starts:
                             [0] 1
                           [1] 0, 1
                          [2] 1, 0, 1
                        [3] 2, 2, 0, 1
                       [4] 5, 5, 3, 0, 1
                    [5] 14, 15, 9, 4, 0, 1
                  [6] 43, 50, 31, 14, 5, 0, 1
              [7] 144, 180, 118, 54, 20, 6, 0, 1
            [8] 523, 695, 481, 229, 85, 27, 7, 0, 1
       [9] 2048, 2869, 2081, 1035, 395, 125, 35, 8, 0, 1

Mathew Englander, Comments on A101494 and A089246, and related sequences

Cf. A047969, A047970, A089302.

First differences by column of A101494.

From Mathew Englander, Feb 25 2021: (Start)
T(n,k) = 0^(n-k) + Sum_{m = k..n-1} C(m,k) * ((n-m)^(m-k) - (n-1-m)^(m-k)).
T(n,k) = Sum_{j = k+1..n-1} C(j,k)*Sum_{i = j..n-1} T(i,j) for 0 <= k < n-1; T(k+1,k)=0 and T(k,k)=1 for k>=0.
G.f. of row n: 1 + Sum_{i = 1..n} (x+n-i)^(i-1)*(x+n-i-1). (End)

Edited and extended by David Wasserman, Sep 07 2005

A112532 First differences of [0, A047970].

Original entry on oeis.org

1, 1, 3, 9, 29, 101, 379, 1525, 6549, 29889, 144419, 736241, 3947725, 22201549, 130624587, 802180701, 5131183301, 34121977865, 235486915507, 1683925343929, 12458499203901, 95237603403381, 751291094637083, 6108883628141189, 51144808472958709, 440444879385258001

Offset: 0

Alford Arnold, Sep 10 2005

nonn

Number of sequences of length n in [n] (endofunctions) whose first run has length equal to the maximum of the sequence.

			The 9 sequences for n=4 (sorted by maximum)
1121,1122,2211,2212, 1113,2223,3331,3332, 4444
The 29 sequences for n=5 (sorted by maximum)
11211,11212,11221,11222, 22111,22112,22121,22122, 11123,11131,11132,11133, 22213,22231,22232,22233, 33311,33312,33313,33321,33322,33323, 11114, 22224, 33334, 44441,44442,44443, 55555

G. C. Greubel, Table of n, a(n) for n = 0..500

Cf. A047970, A112531, A229046.

First differences of column 0 of triangle A089246 (beginning at row 1). With offset 1, first differences of column 0 of triangle A242431. Second differences of column 0 of triangle A101494.

a[n_]:= If[n==0, 1, n + Sum[(i-1)^2*i^(n-i), {i,0,n}]];
Table[a[n], {n, 0, 30}] (* G. C. Greubel, Jan 12 2022 *)

a(n) = n + sum(i = 0, n, (n-i-1)^2 * (n-i)^i); \\ Michel Marcus, Mar 01 2021

[n +sum((j-1)^2*j^(n-j) for j in (0..n)) for n in (0..30)] # G. C. Greubel, Jan 12 2022

G.f.: (1-x)^2*( Sum_{n >= 0} x^n/(1 - (n+2)*x) ). - Peter Bala, Jul 09 2014
From Mathew Englander, Feb 28 2021: (Start)
a(n) = A089246(n+2,0) - A089246(n+1,0).
a(n) = n + Sum_{i = 0..n} (n-i-1)^2 * (n-i)^i. (End)

Corrected by D. S. McNeil, Aug 20 2010

Combinatorial interpretation and examples by Olivier Gérard, Jan 29 2023

A304357 Antidiagonal sums of the first quadrant of array A(k,m) = F_k(m), F_k(m) being the k-th Fibonacci polynomial evaluated at m.

Original entry on oeis.org

0, 1, 1, 3, 5, 13, 32, 94, 297, 1036, 3911, 15918, 69350, 321779, 1582745, 8220349, 44925187, 257563819, 1544896976, 9671289892, 63051738167, 427254561854, 3003872526303, 21876513464296, 164790822258172, 1282198404741305, 10292007232817249, 85126350266370355

Offset: 0

Alois P. Heinz, May 11 2018

nonn

Equivalently, antidiagonal sums of the third quadrant of array A(k,m).

It seems that: a(n+1) is the sum of the n-th antidiagonal of triangle A101494; a(n)-(n mod 2) is the sum of the n-th antidiagonal of array A172236; and a(n+1)+(n mod 2) is the sum of row n of triangle A157103. - Mathew Englander, Feb 28 2021

Alois P. Heinz, Table of n, a(n) for n = 0..600
Wikipedia, Fibonacci polynomials
Wikipedia, Quadrant (plane geometry)

Cf. A000045, A084844, A304359, A101494, A172236, A157103.

F:= (n, k)-> (<<0|1>, <1|k>>^n)[1, 2]:
a:= n-> add(F(j, n-j), j=0..n):
seq(a(n), n=0..30);
# second Maple program:
F:= proc(n, k) option remember;
      `if`(n<2, n, k*F(n-1, k)+F(n-2, k))
    end:
a:= n-> add(F(j, n-j), j=0..n):
seq(a(n), n=0..30);
# third Maple program:
a:= n-> add(combinat[fibonacci](j, n-j), j=0..n):
seq(a(n), n=0..30);

a[n_] := Sum[Fibonacci[j, n - j], {j, 0, n}];
Table[a[n], {n, 0, 30}] (* Jean-François Alcover, Jun 02 2018, from 3rd Maple program *)

a(n) = Sum_{j=0..n} F_j(n-j).

a(n+1) = Sum_{j = 0..n} Sum_{i = j..floor((n+j)/2)} binomial(i,j)*(n+j-2*i)^j (empirically). - Mathew Englander, Feb 28 2021

A152798 Triangle defined by T(n,k) = Sum_{j=0..k} C(k,j)*T(n-1,j+k) for n>k>0 with T(n,0)=T(n,n)=1, read by rows.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 3, 3, 1, 1, 1, 6, 6, 4, 1, 1, 1, 12, 15, 10, 5, 1, 1, 1, 27, 40, 29, 15, 6, 1, 1, 1, 67, 113, 93, 49, 21, 7, 1, 1, 1, 180, 348, 310, 180, 76, 28, 8, 1, 1, 1, 528, 1148, 1106, 685, 311, 111, 36, 9, 1, 1, 1, 1676, 4045, 4205, 2748, 1322, 497, 155, 45

Offset: 0

Paul D. Hanna, Dec 23 2008

			Triangle begins:
1;
1, 1;
1, 1, 1;
1, 2, 1, 1;
1, 3, 3, 1, 1;
1, 6, 6, 4, 1, 1;
1, 12, 15, 10, 5, 1, 1;
1, 27, 40, 29, 15, 6, 1, 1;
1, 67, 113, 93, 49, 21, 7, 1, 1;
1, 180, 348, 310, 180, 76, 28, 8, 1, 1;
1, 528, 1148, 1106, 685, 311, 111, 36, 9, 1, 1;
1, 1676, 4045, 4205, 2748, 1322, 497, 155, 45, 10, 1, 1;
1, 5721, 15203, 16912, 11683, 5858, 2323, 750, 209, 55, 11, 1, 1;
1, 20924, 60710, 71858, 52262, 27349, 11230, 3809, 1083, 274, 66, 12, 1, 1; ...
ILLUSTRATE RECURRENCE:
T(6,1) = T(5,1) + T(5,2) = 6 + 6 = 12;
T(7,2) = T(6,2) + 2*T(6,3) + T(6,4) = 6 + 2*4 + 1 = 15;
T(8,3) = T(7,3) + 3*T(7,4) + 3*T(7,5) + T(7,6) = 29 + 3*15 + 3*6 + 1 = 93.
Note that column 1 equals A122889: [1,1,2,3,6,12,27,67,180,528,...]
which is the antidiagonal sums of triangle A122888.
RELATED TRIANGLE A122888 begins:
1;
1, 1;
1, 2, 2, 1;
1, 3, 6, 9, 10, 8, 4, 1;
1, 4, 12, 30, 64, 118, 188, 258, 302, 298, 244, 162, 84, 32, 8, 1;
1, 5, 20, 70, 220, 630, 1656, 4014, 8994, 18654, 35832, 63750,...;
1, 6, 30, 135, 560, 2170, 7916, 27326, 89582, 279622, 832680,...;
1, 7, 42, 231, 1190, 5810, 27076, 121023, 520626, 2161158,...;
1, 8, 56, 364, 2240, 13188, 74760, 409836, 2179556, 11271436,...; ...
in which the g.f. of row n equals the n-th iteration of (x+x^2).

Cf. A122888; columns: A122889, A152799; variant: A101494.

PARI
```
T(n, k)=if(n
				
```

A242431 Triangle read by rows: T(n, k) = (k + 1)*T(n-1, k) + Sum_{j=k..n-1} T(n-1, j) for k < n, T(n, n) = 1. T(n, k) for n >= 0 and 0 <= k <= n.

Original entry on oeis.org

1, 2, 1, 5, 3, 1, 14, 10, 4, 1, 43, 35, 17, 5, 1, 144, 128, 74, 26, 6, 1, 523, 491, 329, 137, 37, 7, 1, 2048, 1984, 1498, 730, 230, 50, 8, 1, 8597, 8469, 7011, 3939, 1439, 359, 65, 9, 1, 38486, 38230, 33856, 21568, 9068, 2588, 530, 82, 10, 1

Offset: 0

Peter Luschny, May 14 2014

			0|    1;
1|    2,    1;
2|    5,    3,    1;
3|   14,   10,    4,   1;
4|   43,   35,   17,   5,   1;
5|  144,  128,   74,  26,   6,  1;
6|  523,  491,  329, 137,  37,  7, 1;
7| 2048, 1984, 1498, 730, 230, 50, 8, 1;

Peter Luschny, Rows n = 0..50, flattened.
Mathew Englander, Comments on A101494 and A089246, and related sequences

Cf. A003101, A026898, A047969, A047970, A101494, A089246.

T := proc(n, k) option remember; local j;
    if k=n then 1
  elif k>n then 0
  else (k+1)*T(n-1, k) + add(T(n-1, j), j=k..n)
    fi end:
seq(print(seq(T(n,k), k=0..n)), n=0..7);

def A242431_rows():
    T = []; n = 0
    while True:
        T.append(1)
        yield T
        for k in (0..n):
            T[k] = (k+1)*T[k] + add(T[j] for j in (k..n))
        n += 1
a = A242431_rows()
for n in range(8): next(a)

T(n, 0) = A047970(n).
Sum_{k=0..n} T(n, k) = A112532(n+1).
From Mathew Englander, Feb 25 2021: (Start)
T(n,k) = 1 + Sum_{i = k+1..n} i*(i+1)^(n-i).
T(n,k) = T(n,k+1) + (k+1)*(k+2)^(n-k-1) for 0 <= k < n.
T(n,k) = T(n,k+1) + (k+2)*(T(n-1,k) - T(n-1,k+1)) for 0 <= k <= n-2.
T(n,k) = Sum_{i = 0..n-k} (k+2)^i*A089246(n-k,i).
Sum_{i = k..n} T(i,k) = Sum_{i = 0..n-k} (n+2-i)^i = Sum_{i = 0..n-k} A101494(n-k,i)*(k+2)^i. (End)

cp's OEIS Frontend

A101495 Column 1 of triangle A101494.

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A026898 a(n) = Sum_{k=0..n} (n-k+1)^k.

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A047970 Antidiagonal sums of nexus numbers (A047969).

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A089246 Triangular array read by rows: a(n, k) is the number of ordered m-tuples of positive integers (x_1, ..., x_m) such that max x_i = n+1-m and there are k ones (0 <= k <= n).

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A112532 First differences of [0, A047970].

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A304357 Antidiagonal sums of the first quadrant of array A(k,m) = F_k(m), F_k(m) being the k-th Fibonacci polynomial evaluated at m.

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