A000994 -id:A000994 - cp's OEIS frontend

A051139 a(n) = A000994(n+2) - A000995(n+2).

1, 0, 0, 1, 3, 7, 19, 64, 236, 893, 3489, 14347, 62429, 285580, 1362372, 6749337, 34683211, 184774851, 1019351071, 5813020060, 34206071992, 207391313189, 1294092695729, 8302365433863, 54713262186745, 370027343460584, 2565874205681368

Offset: 0

N. J. A. Sloane

			a(7) = 359 - 295 = 64.

Reinhard Zumkeller, Table of n, a(n) for n = 0..100

Cf. A000994, A000995.

a051139 n = a000994 (n + 2) - a000995 (n + 2)
-- Reinhard Zumkeller, Jun 02 2013

a4[?Negative] = 0; a4[n] := a4[n] = 1 + Sum[ Binomial[n, k]*a4[k-2], {k, 2, n}]; a5[?Negative] = 0; a5[n] := a5[n] = n + Sum[ Binomial[n, k]*a5[k-2], {k, 2, n}]; a[n_] := a4[n] - a5[n]; Table[a[n], {n, 0, 26}] (* Jean-François Alcover, Oct 11 2013 *)

For recurrence see A000994 and A000995.

A051140 a(n) = (A000110(n) - A000994(n+2))/2.

Original entry on oeis.org

0, 0, 0, 0, 1, 8, 47, 259, 1437, 8208, 48659, 300553, 1937958, 13052028, 91784452, 673328720, 5146726651, 40934788112, 338292605257, 2900716040495, 25769979503573, 236876549095324, 2249873611378525, 22053647698650293, 222832106687369092

Offset: 0

N. J. A. Sloane

nonn

			a(7) = (877 - 359)/2 = 259.

Cf. A000110, A000994.

A007476 Shifts 2 places left under binomial transform.

Original entry on oeis.org

1, 1, 1, 2, 4, 9, 23, 65, 199, 654, 2296, 8569, 33825, 140581, 612933, 2795182, 13298464, 65852873, 338694479, 1805812309, 9963840219, 56807228074, 334192384460, 2026044619017, 12642938684817, 81118550133657, 534598577947465, 3615474317688778, 25070063421597484

Offset: 0

N. J. A. Sloane

Starting (1, 2, 4, 9, 23, ...) = row sums of triangle A153859. - Gary W. Adamson, Jan 02 2009
Binomial transform of the sequence starting (1, 1, 2, 4, 9, ...) = first differences of (1, 2, 4, 9, 23, ...); that is, (1, 2, 5, 14, 42, 134, 455, 1642, ...). - Gary W. Adamson, May 20 2013
Row sums of triangle A256161. - Margaret A. Readdy, Mar 16 2015
RG-words corresponding to set partitions of {1, ..., n} with every even entry appearing exactly once. - Margaret A. Readdy, Mar 16 2015
a(n) is the number of partitions of [n] whose blocks can be written such that the smallest elements form an increasing sequence and the largest elements form a decreasing sequence. a(5) = 9: 12345, 1235|4, 1245|3, 125|34, 1345|2, 135|24, 145|23, 15|234, 15|24|3. - Alois P. Heinz, Apr 24 2023

N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Alois P. Heinz, Table of n, a(n) for n = 0..650 (first 101 terms from T. D. Noe)
M. Bernstein and N. J. A. Sloane, Some canonical sequences of integers, arXiv:math/0205301 [math.CO], 2002; Linear Alg. Applications, 226-228 (1995), 57-72; erratum 320 (2000), 210.
M. Bernstein and N. J. A. Sloane, Some canonical sequences of integers, Linear Alg. Applications, 226-228 (1995), 57-72; erratum 320 (2000), 210. [Link to Lin. Alg. Applic. version together with omitted figures]
Yue Cai and Margaret Readdy, Negative q-Stirling numbers, arXiv:1506.03249 [math.CO], 2015.
A. Claesson and T. Mansour, Permutations avoiding a pair of Babson-Steingrimsson patterns, arXiv:math/0107044 [math.CO], 2001-2010.
Rigoberto Flórez, José L. Ramírez, Fabio A. Velandia, and Diego Villamizar, Some Connections Between Restricted Dyck Paths, Polyominoes, and Non-Crossing Partitions, arXiv:2308.02059 [math.CO], 2023. See Table 1 p. 13.
N. J. A. Sloane, Transforms
L. Sze, OEIS conjecture 70

Cf. A000994, A000995, A092920, A153859.

Row sums of A246118.

a:= proc(n) option remember; `if`(n<2, 1,
      add(a(j)*binomial(n-2, j), j=0..n-2))
    end:
seq(a(n), n=0..31);  # Alois P. Heinz, Jul 29 2019

a[0] = a[1] = 1; a[n_] := a[n] = Sum[Binomial[n-2, k] a[k], {k, 0, n-2}]; Table[a[n], {n, 0, 24}] (* Jean-François Alcover, Aug 08 2012, after Ralf Stephan *)

a(n)=if(n<2, 1, sum(k=0, n-2, binomial(n-2, k)*a(k))) /* Ralf Stephan; corrected by Manuel Blum, May 22 2010 */

G.f.: Sum_{k>=0} x^(2k)/(Product_{m=0..k-1} (1-mx) * Product_{m=0..k+1} (1-mx)).
G.f. A(x) satisfies A(x) = 1 + x + (x^2/(1-x))*A(x/(1-x)). - Vladimir Kruchinin, Nov 28 2011
a(n) = A000994(n) + A000995(n). - Peter Bala, Jan 27 2015

Spelling correction by Jason G. Wurtzel, Aug 22 2010

A000995 Shifts left two terms under the binomial transform.

Original entry on oeis.org

0, 1, 0, 1, 2, 4, 10, 29, 90, 295, 1030, 3838, 15168, 63117, 275252, 1254801, 5968046, 29551768, 152005634, 810518729, 4472244574, 25497104007, 149993156234, 909326652914, 5674422994544, 36408092349897, 239942657880360

Offset: 0

N. J. A. Sloane

The binomial transform of A000995 has g.f. x*c(x)^2/(1+x^2*c(x)^2). - Paul Barry, Oct 06 2007
Equals row sums of triangle A137854 such that A000995(3) = 1 = first row of triangle A137854. - Gary W. Adamson, Feb 15 2008
a(n) is the number of permutations of [n-1] that avoid both of the dashed patterns 1-23 and 3-12 and start with an ascent (or are empty). For example, a(5)=4 counts 1432, 2314, 2431, 3421. - David Callan, Dec 02 2011

			A(x) = x + x^3/(1-x)^2 + x^5/((1-x)*(1-2x))^2 + x^7/((1-x)*(1-2x)*(1-3x))^2 +...

Ulrike Sattler, Decidable classes of formal power series with nice closure properties, Diplomarbeit im Fach Informatik, Univ. Erlangen - Nuernberg, Jul 27 1994
N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

T. D. Noe, Table of n, a(n) for n = 0..100
S. Tauber, On generalizations of the exponential function, Amer. Math. Monthly, 67 (1960), 763-767.

Cf. A000994, A051139, A051140.
Cf. A137854.
Cf. A007318.

a000995 n = a000995_list !! n
a000995_list = 0 : 1 : vs where
  vs = 0 : 1 : g 2 where
    g x = (x + sum (zipWith (*) (map (a007318' x) [2..x]) vs)) : g (x + 1)
-- Reinhard Zumkeller, Jun 02 2013

A000995 := proc(n) local k; option remember; if n <= 1 then n else n + add(binomial(n, k)*A000995(k - 2), k = 2 .. n); fi; end;

a[n_] := a[n] = If[n <= 1, n, n + Sum[Binomial[n, k]*a[k-2], {k, 2, n}]]; Join[{0, 1}, Table[a[n], {n, 0, 24}]] (* Jean-François Alcover, May 18 2011, after Maple prog. *)
(* Computation using e.g.f.: *)
nn=20; S=(Series[-2 E^(t/2) Sqrt[E^ t] (BesselI[0, 2] BesselK[0, 2 Sqrt[E^t]] - BesselK[0, 2] Hypergeometric0F1[1, E^t]), {t, 0, nn}]); Flatten[{0, 1, FullSimplify[Table[CoefficientList[Normal[S], t][[i]] (i - 1)!, {i, 1, nn}]]}] (* Pierre-Louis Giscard, Aug 12 2014 *)

a(n)=polcoeff(sum(k=0,n,x^(2*k+1)/prod(j=0,k,1-j*x+x*O(x^n))^2),n) \\ Paul D. Hanna, Oct 28 2006

Since this satisfies a recurrence similar to that of the Bell numbers (A000110), the asymptotic behavior is presumably just as complicated - see A000110 for details.
However, A000994(n)/A000995(n) [ e.g., 77464/63117 ] -> 1.228..., the constant in A051148 and A051149.
O.g.f.: A(x) = Sum_{n>=0} x^(2*n+1)/Product_{k=0..n} (1-k*x)^2. - Paul D. Hanna, Oct 28 2006
G.f.: (1+2*x^2*c(x)^2)/(1+x^2*c(x^2)), c(x) the g.f. of A000108. - Paul Barry, Oct 06 2007. This g.f. is incorrect. - Vaclav Kotesovec, Aug 14 2014
E.g.f: -2 * exp(x) *( BesselI_0(2) * BesselK_0(2*exp(x/2)) - BesselK_0(2) * 0F1([], [1], exp(x)) ); see the Mathematica program. - Pierre-Louis Giscard, Aug 12 2014
G.f. A(x) satisfies: A(x) = x*(1 + x*A(x/(1 - x))/(1 - x)). - Ilya Gutkovskiy, May 02 2019

More terms from Paul D. Hanna, Oct 28 2006

A086880 a(n) = floor( sum(k=0, infinity, k^n/(k!)^2 ) ); related to generalized Bell numbers.

Original entry on oeis.org

2, 1, 2, 3, 7, 17, 45, 128, 391, 1287, 4524, 16889, 66657, 276982, 1207598, 5507362, 26203307, 129757596, 667358910, 3558097578, 19632277761, 111930731957, 658482495614, 3992062349412, 24911272290567, 159833355923362

Offset: 0

Paul D. Hanna, Sep 16 2003

nonn

Define B(n) = sum(k=0, infinity, k^n/(k!)^2), then there exists a complex linear relation: B(3) = B(2) + B(1); B(4) = 2*B(3); B(5) = 2*B(4) + B(2); B(6) = 5*B(4) + 3*B(2); B(7) = 7*B(5) + B(3); B(12) = B(11) + 11*B(10); ...

			a(5) = floor(1^5/(1!)^2 + 2^5/(2!)^2 + 3^5/(3!)^2 + 4^5/(4!)^2 +...)

Vaclav Kotesovec, Table of n, a(n) for n = 0..646

Cf. A000994, A000995, A006789.

Table[Floor[Sum[k^n/(k!)^2,{k,0,Infinity}]],{n,0,20}] (* Vaclav Kotesovec, Jul 31 2014 *)
Flatten[{2, 1, Table[Floor[HypergeometricPFQ[ConstantArray[2, n-2], ConstantArray[1, n-1], 1]], {n,2,20}]}] (* Vaclav Kotesovec, May 23 2015 *)

sum(k>=0, k^n/(k!)^2) = A000994(n)*BesselI(0, 2) + A000995(n)*BesselI(1, 2), using Bessel function values BesselI(0, 2)=2.2795853023..., BesselI(1, 2) = 1.5906368546... (A096789) and where A000994 and A000995 shift 2 places left under binomial transform: A000994={1, 0, 1, 1, 2, 5, 13, 36, 109, 359, 1266, 4731, ...} A000995={0, 1, 0, 1, 2, 4, 10, 29, 90, 295, 1030, 3838, ...}.

A143983 Triangle T(n,k), n>=1, 1<=k<=n, read by rows, where sequence a_k of column k has a_k(0)=1, followed by (k-1)-fold 0 and a_k(n) shifts k places down under binomial transform.

Original entry on oeis.org

1, 2, 1, 5, 1, 1, 15, 2, 1, 1, 52, 5, 1, 1, 1, 203, 13, 2, 1, 1, 1, 877, 36, 6, 1, 1, 1, 1, 4140, 109, 17, 2, 1, 1, 1, 1, 21147, 359, 44, 7, 1, 1, 1, 1, 1, 115975, 1266, 112, 23, 2, 1, 1, 1, 1, 1, 678570, 4731, 304, 65, 8, 1, 1, 1, 1, 1, 1, 4213597, 18657, 918, 165, 30, 2, 1, 1, 1, 1, 1, 1

Offset: 1

Alois P. Heinz, Sep 06 2008

The matrix inverse starts:
     1;
    -2,   1;
    -3,  -1,  1;
    -8,  -1, -1,  1;
   -31,  -3,  0, -1,  1;
  -132,  -7, -1,  0, -1,  1;
  -616, -19, -4,  0,  0, -1, 1; - R. J. Mathar, Mar 22 2013

			T(5,2) = 5, because [1,3,3,1] * [1,0,1,1] = 5.
Triangle begins:
:      1;
:      2,    1;
:      5,    1,   1;
:     15,    2,   1,  1;
:     52,    5,   1,  1, 1;
:    203,   13,   2,  1, 1, 1;
:    877,   36,   6,  1, 1, 1, 1;
:   4140,  109,  17,  2, 1, 1, 1, 1;
:  21147,  359,  44,  7, 1, 1, 1, 1, 1;
: 115975, 1266, 112, 23, 2, 1, 1, 1, 1, 1;

Alois P. Heinz, Rows n = 1..141, flattened
N. J. A. Sloane, Transforms

Columns 1-6 give: A000110, A000994, A000996, A010748, A010749, A010750.

Cf. A007318.

T:= proc(n, k) option remember; `if`(n

t[n_, k_] := t[n, k] = If[n < k, If[n == 0, 1, 0], Sum[Binomial[n-k, j]*t[j, k], {j, 0, n-k}]]; Table[Table[t[n, k], {k, 1, n}], {n, 1, 13}] // Flatten (* Jean-François Alcover, Dec 18 2013, translated from Maple *)

T(n,k) = Sum_{j=0..n-k} C(n-k,j)*T(j,k) if n>=k, else T(n,k) = 1 if n=1, else T(n,k) = 0.

A265165 a(n) = sum of the n-th column of the array A265163(n,k). See Comments for more details.

Original entry on oeis.org

1, 0, 1, 2, 7, 32, 179, 1182, 8993, 77440, 744425, 7901410, 91774375, 1157782560, 15764338315, 230416499390, 3598316747905, 59792454064640, 1053360827319185, 19610513077334850, 384703418451703175, 7931544941793536800, 171459202078545968675, 3877969156687438765150

Offset: 0

Cyril Banderier, Dec 07 2015; revised Feb 06 2017

nonn

A right-jump in a permutation consists of taking an element and moving it somewhere to its right.
The set P(k) of permutations reachable from the identity after at most k right-jumps is a permutation-pattern avoiding set: it coincides with the set of permutation avoiding a set of patterns.
We define B(k) to be the smallest such set of "forbidden patterns" (the permutation pattern community calls such a set a "basis" for P(k), and its elements can be referred to as "right-jump basis permutations").
The number b(n,k) of permutations of size n in B(k) is given by the array A265163.
The row sums give the sequence A265164 (i.e. this counts the permutations of any size in the basis B(k)).
The column sums give the present sequence (i.e. this counts the permutations of size n in any B(k)).

			G.f. = x^2 + 2*x^3 + 7*x^4 + 32*x^5 + 179*x^6 + 1182*x^7 + 8993*x^8 + ...
The basis permutations of size 2 are 21 thus a(2)=1.
The basis permutations of size 3 are 312 and 321 thus a(3)=2.
The basis permutations of size 4 are 2143, 4123, 4132, 4213, 4231, 4312, 4321, thus a(4)=7.

Alois P. Heinz, Table of n, a(n) for n = 0..450
Cyril Banderier, Jean-Luc Baril, Céline Moreira Dos Santos, Right jumps in permutations, Permutation Patterns 2015.
Cyril Banderier, Florian Luca, On the period mod m of polynomially-recursive sequences: a case study, arXiv:1903.01986 [math.NT], 2019.

Cf. A000166, A000994, A001040, A265163, A265164.

gfun[rectoproc]({(n^2+3*n+1)*a(n)+(-2*n-4)*a(n+1)+a(n+2), a(0)=0, a(1)=0, a(2)=1}, a(n), remember);
# second Maple program:
a:= proc(n) option remember; `if`(n=0, 1, add(
      a(n-j)*binomial(n-2, j-2)*(j-1)!, j=2..n))
    end:
seq(a(n), n=0..23);  # Alois P. Heinz, Jul 03 2023

a[ n_] := If[ n < 1, 0, With[ {w = (1 + Sqrt[5])/2}, n! SeriesCoefficient[ w (1 - x)^(1 - w) - (1 - w) (1 - x)^w, {x, 0, n}]/Sqrt[5] // Simplify]]; (* Michael Somos, Jan 27 2017 *)
RecurrenceTable[{a[n+2] == 2 n*a[n+1] - (n^2 - n - 1)*a[n], a[1] == 0, a[2] == 1}, a, {n, 1, 25}] (* Vaclav Kotesovec, Jan 20 2019 *)

{a(n) = my(A); if( n<3, n==2, A = vector(n); A[2] = 1; for(k = 1, n-2, A[k + 2] = 2*k*A[k + 1] - (k^2 - k - 1)*A[k]); A[n])}; /* Michael Somos, Jan 27 2017 */

{a(n) = my(w); if( n<1, 0, w = quadgen(5); n! * polcoeff( imag( w * (1 - x + x * O(x^n))^(1 - w) ), n))}; /* Michael Somos, Jan 27 2017 */

a(n+2) = 2n*a(n+1) - (n^2-n-1)*a(n) if n>0.
E.g.f.: -1 + (w * (1 - x)^(1 - w) - (1 - w) * (1 - x)^w) / sqrt(5) where w = (1 + sqrt(5))/2. - Michael Somos, Jan 27 2017
E.g.f. A(x) satisfies 0 == 1 + A(x) - (1 - x)^2 * A''(x). - Michael Somos, Jan 27 2017
0 = a(n)*(+4*a(n+1) + 2*a(n+2) - 6*a(n+3) + a(n+4)) + a(n+1)*(+4*a(n+1) + 6*a(n+2) - 4*a(n+3)) + a(n+2)*(+3*a(n+2)) if n>0. - Michael Somos, Jan 27 2017
a(n) ~ n! * (1 + 1/sqrt(5)) / (2 * Gamma((sqrt(5)-1)/2) * n^((3-sqrt(5))/2)). - Vaclav Kotesovec, Jan 20 2019
a(n) = (-1)^(n+1) * Sum_{i=1..n+1} A008275(n+1,i) * A001519(i-1). - Max Alekseyev, Dec 05 2020

a(0)=1 prepended by Alois P. Heinz, Jul 03 2023

A351143 G.f. A(x) satisfies: A(x) = 1 + x^2 * A(x/(1 - 2x)) / (1 - 2x).

Original entry on oeis.org

1, 0, 1, 2, 5, 16, 61, 258, 1177, 5776, 30537, 173394, 1050045, 6732608, 45459493, 322141106, 2390075249, 18525967328, 149684238801, 1257802518754, 10969260208565, 99100423076912, 926030783479629, 8937741026924450, 88988433270106249, 912906193294355952

Offset: 0

Ilya Gutkovskiy, Feb 02 2022

nonn

Shifts 2 places left under 2nd-order binomial transform.

Cf. A000994, A004211, A007472, A351144, A351150, A351151, A351152.

bintr:= proc(p) local b;
          b:= proc(n) option remember; add(p(k)*binomial(n, k), k=0..n) end
        end:
b:= (bintr@@2)(a):
a:= n-> `if`(n<2, 1-n, b(n-2)):
seq(a(n), n=0..25);  # Alois P. Heinz, Apr 07 2025

nmax = 25; A[] = 0; Do[A[x] = 1 + x^2 A[x/(1 - 2 x)]/(1 - 2 x) + O[x]^(nmax + 1) // Normal, nmax + 1]; CoefficientList[A[x], x]
a[0] = 1; a[1] = 0; a[n_] := a[n] = Sum[Binomial[n - 2, k] 2^k a[n - k - 2], {k, 0, n - 2}]; Table[a[n], {n, 0, 25}]
(* another program *)
B[x_] := BesselK[1, 1]*BesselI[0, Exp[x]] + BesselI[1, 1]*BesselK[0, Exp[x]];
a[n_] := SeriesCoefficient[FullSimplify[Series[B[x], {x, 0, n}]], n] n!
Table[a[n], {n, 0, 30}] (* Ven Popov, Apr 25 2025 *)

a(0) = 1, a(1) = 0; a(n) = Sum_{k=0..n-2} binomial(n-2,k) * 2^k * a(n-k-2).

E.g.f.: BesselK(1, 1)*BesselI(0, exp(x)) + BesselI(1, 1)*BesselK(0, exp(x)). - Ven Popov, Apr 25 2025

A351144 G.f. A(x) satisfies: A(x) = 1 + x^2 * A(x/(1 - 3x)) / (1 - 3x).

Original entry on oeis.org

1, 0, 1, 3, 10, 39, 181, 972, 5797, 37389, 258202, 1905681, 15016465, 125920872, 1117950913, 10452866439, 102485649754, 1050464300187, 11231883627301, 125055844922916, 1447371528438565, 17382103226123313, 216221862096537994, 2781342531957176085, 36942930754308211969

Offset: 0

Ilya Gutkovskiy, Feb 02 2022

nonn

Shifts 2 places left under 3rd-order binomial transform.

Cf. A000994, A004212, A351049, A351143, A351150, A351151, A351152.

nmax = 24; A[] = 0; Do[A[x] = 1 + x^2 A[x/(1 - 3 x)]/(1 - 3 x) + O[x]^(nmax + 1) // Normal, nmax + 1]; CoefficientList[A[x], x]
a[0] = 1; a[1] = 0; a[n_] := a[n] = Sum[Binomial[n - 2, k] 3^k a[n - k - 2], {k, 0, n - 2}]; Table[a[n], {n, 0, 24}]

a(0) = 1, a(1) = 0; a(n) = Sum_{k=0..n-2} binomial(n-2,k) * 3^k * a(n-k-2).

A351150 G.f. A(x) satisfies: A(x) = 1 + x^2 * A(x/(1 - 4x)) / (1 - 4x).

Original entry on oeis.org

1, 0, 1, 4, 17, 80, 433, 2724, 19489, 153536, 1301601, 11754084, 112802097, 1150079056, 12437130001, 142144768324, 1709041379393, 21522252928000, 282920962675905, 3873124754702660, 55125031662585425, 814541756986322128, 12477752083406752881, 197861190429889969252

Offset: 0

Ilya Gutkovskiy, Feb 02 2022

nonn

Shifts 2 places left under 4th-order binomial transform.

Cf. A000994, A004213, A351050, A351143, A351144, A351151, A351152.

nmax = 23; A[] = 0; Do[A[x] = 1 + x^2 A[x/(1 - 4 x)]/(1 - 4 x) + O[x]^(nmax + 1) // Normal, nmax + 1]; CoefficientList[A[x], x]
a[0] = 1; a[1] = 0; a[n_] := a[n] = Sum[Binomial[n - 2, k] 4^k a[n - k - 2], {k, 0, n - 2}]; Table[a[n], {n, 0, 23}]

a(0) = 1, a(1) = 0; a(n) = Sum_{k=0..n-2} binomial(n-2,k) * 4^k * a(n-k-2).

cp's OEIS Frontend

A051139 a(n) = A000994(n+2) - A000995(n+2).

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A051140 a(n) = (A000110(n) - A000994(n+2))/2.

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A007476 Shifts 2 places left under binomial transform.

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A000995 Shifts left two terms under the binomial transform.

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A086880 a(n) = floor( sum(k=0, infinity, k^n/(k!)^2 ) ); related to generalized Bell numbers.

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A143983 Triangle T(n,k), n>=1, 1<=k<=n, read by rows, where sequence a_k of column k has a_k(0)=1, followed by (k-1)-fold 0 and a_k(n) shifts k places down under binomial transform.

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A265165 a(n) = sum of the n-th column of the array A265163(n,k). See Comments for more details.

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A351143 G.f. A(x) satisfies: A(x) = 1 + x^2 * A(x/(1 - 2x)) / (1 - 2x).

A351144 G.f. A(x) satisfies: A(x) = 1 + x^2 * A(x/(1 - 3x)) / (1 - 3x).

A351150 G.f. A(x) satisfies: A(x) = 1 + x^2 * A(x/(1 - 4x)) / (1 - 4x).