A007476 -id:A007476 - cp's OEIS frontend

A153859 Triangle read by rows, A007318 * (A007476 * 0 ^(n-k)).

1, 1, 1, 1, 2, 1, 1, 3, 3, 2, 1, 4, 6, 8, 4, 1, 5, 10, 20, 20, 9, 1, 6, 15, 40, 60, 54, 23, 1, 7, 21, 70, 140, 189, 161, 65, 1, 8, 28, 112, 280, 504, 644, 520, 199, 1, 9, 36, 168, 504, 1134, 1932, 2340, 1791, 654, 1, 10, 45, 240, 840, 2268, 4830, 7800, 8955, 6540, 2296

Offset: 0

Gary W. Adamson, Jan 02 2009

Row sums = A007476 starting (1, 2, 4, 9, 23, 65, 199,...).

			First few rows of the triangle =
1;
1, 1;
1, 2, 1;
1, 3, 3, 2;
1, 4, 6, 8, 4;
1, 5, 10, 20, 20, 9;
1, 6, 15, 40, 60, 54, 23;
1, 7, 21, 70, 140, 189, 161, 65;
1, 8, 28, 112, 280, 504, 644, 520, 199;
1, 9, 36, 168, 504, 1134, 1932, 2340, 1791, 654;
1, 10, 45, 240, 840, 2268, 4830, 7800, 8955, 6540, 2296;
...

Cf. A007476

Triangle read by rows A007318 * (A007476 * 0^(n-k)) = binomial transform of an infinite lower triangular matrix with A007476 as the main diagonal: (1, 1, 1, 2, 4, 9, 23, 65, 199,...) and the rest zeros.

A000994 Shifts 2 places left under binomial transform.

Original entry on oeis.org

1, 0, 1, 1, 2, 5, 13, 36, 109, 359, 1266, 4731, 18657, 77464, 337681, 1540381, 7330418, 36301105, 186688845, 995293580, 5491595645, 31310124067, 184199228226, 1116717966103, 6968515690273, 44710457783760, 294655920067105, 1992750830574681, 13817968813639426

Offset: 0

N. J. A. Sloane

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 a descent (or are a singleton). For example, a(5)=5 counts 2143, 3142, 3214, 3241, 4321. - David Callan, Nov 21 2011

			A(x) = 1 + x^2/(1-x) + x^4/((1-x)^2*(1-2x)) + x^6/((1-x)^2*(1-2x)^2*(1-3x)) +...

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).

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, Linear Alg. Applications, 226-228 (1995), 57-72; erratum 320 (2000), 210; arXiv:math/0205301 [math.CO], 2002.
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]
R. K. Guy, The Second Strong Law of Small Numbers, Math. Mag, 63 (1990), no. 1, 3-20. [Annotated scanned copy]
N. J. A. Sloane, Transforms
S. Tauber, On generalizations of the exponential function, Amer. Math. Monthly, 67 (1960), 763-767.

Cf. A000995, A051139, A051140. Cf. A007318.

Column k=2 of A143983. Cf. A007476, A088022, A086880.

a000994 n = a000994_list !! n
a000994_list = 1 : 0 : us where
  us = 1 : 1 : f 2 where
    f x = (1 + sum (zipWith (*) (map (a007318' x) [2..x]) us)) : f (x + 1)
-- Reinhard Zumkeller, Jun 02 2013

A000994 := proc(n) local k; option remember; if n <= 1 then 1 else 1 + add(binomial(n, k)*A000994(k - 2), k = 2 .. n); fi; end;

a[n_] := a[n] = 1 + Sum[Binomial[n, k]*a[k-2], {k, 2, n}]; Join[{1, 0}, Table[a[n], {n, 0, 24}]] (* Jean-François Alcover, Oct 11 2011, after Maple *)

a(n)=polcoeff(sum(k=0, n, x^(2*k)*(1-k*x)/prod(j=0, k, 1-j*x+x*O(x^n))^2), n) \\ Paul D. Hanna, Nov 02 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, a(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-n*x)/Product_{k=0..n} (1-k*x)^2. - Paul D. Hanna, Nov 02 2006
Let S(n) = Sum_{k >= 1} k^n/k!^2. Then S(n) = a(n)*S(0) + A000995(n)*S(1) is stated in A086880, where S(0) = 2.279585302... (see A070910) and S(1) = 1.590636854... (see A096789). Cf. A088022. - Peter Bala, Jan 27 2015
G.f. A(x) satisfies: A(x) = 1 + x^2 * A(x/(1 - x)) / (1 - x). - Ilya Gutkovskiy, Aug 09 2020

A351049 G.f. A(x) satisfies: A(x) = 1 + x + x^2 * A(x/(1 - 3x)) / (1 - 3x).

Original entry on oeis.org

1, 1, 1, 4, 16, 67, 307, 1585, 9235, 59548, 415564, 3094807, 24452785, 204611653, 1810429597, 16892405896, 165592138372, 1698918207403, 18184602679435, 202577753111653, 2344503929765023, 28146188358379120, 349996346545057288, 4501360727764475503

Offset: 0

Ilya Gutkovskiy, Jan 30 2022

nonn

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

Seiichi Manyama, Table of n, a(n) for n = 0..540

Cf. A004212, A007472, A007476, A351050.

nmax = 23; A[] = 0; Do[A[x] = 1 + x + x^2 A[x/(1 - 3 x)]/(1 - 3 x) + O[x]^(nmax + 1) // Normal, nmax + 1]; CoefficientList[A[x], x]
a[0] = a[1] = 1; a[n_] := a[n] = Sum[Binomial[n - 2, k] 3^k a[n - k - 2], {k, 0, n - 2}]; Table[a[n], {n, 0, 23}]

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

A351050 G.f. A(x) satisfies: A(x) = 1 + x + x^2 * A(x/(1 - 4x)) / (1 - 4x).

Original entry on oeis.org

1, 1, 1, 5, 25, 129, 713, 4373, 30289, 235041, 1998001, 18226117, 176364969, 1803064033, 19463340729, 221691818005, 2658751147297, 33458500940993, 440140082161121, 6032572875160069, 85936355674437561, 1270176766188103105, 19453176663852208937

Offset: 0

Ilya Gutkovskiy, Jan 30 2022

nonn

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

Seiichi Manyama, Table of n, a(n) for n = 0..516

Cf. A004213, A007472, A007476, A351049.

nmax = 22; A[] = 0; Do[A[x] = 1 + x + x^2 A[x/(1 - 4 x)]/(1 - 4 x) + O[x]^(nmax + 1) // Normal, nmax + 1]; CoefficientList[A[x], x]
a[0] = a[1] = 1; a[n_] := a[n] = Sum[Binomial[n - 2, k] 4^k a[n - k - 2], {k, 0, n - 2}]; Table[a[n], {n, 0, 22}]

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

A351437 G.f. A(x) satisfies: A(x) = 1 + x + x^2 * A(x/(1 - x)) / (1 - x)^2.

Original entry on oeis.org

1, 1, 1, 3, 7, 17, 47, 145, 481, 1691, 6295, 24805, 103095, 449805, 2052081, 9762699, 48334855, 248568321, 1325297879, 7312927481, 41694974649, 245288605059, 1487041552343, 9279329735685, 59537092965663, 392371097100373, 2653606218921673, 18400405626141667, 130712743774279015

Offset: 0

Ilya Gutkovskiy, Feb 11 2022

nonn

Cf. A007476, A040027, A351438.

nmax = 28; A[] = 0; Do[A[x] = 1 + x + x^2 A[x/(1 - x)]/(1 - x)^2 + O[x]^(nmax + 1) // Normal, nmax + 1]; CoefficientList[A[x], x]
a[0] = a[1] = 1; a[n_] := a[n] = Sum[Binomial[n - 1, k + 1] a[k], {k, 0, n - 2}]; Table[a[n], {n, 0, 28}]

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

A210545 T(n,k) = number of arrays of n nonnegative integers with value i>0 appearing only after i-1 has appeared at least k times.

Original entry on oeis.org

1, 1, 2, 1, 1, 5, 1, 1, 2, 15, 1, 1, 1, 4, 52, 1, 1, 1, 2, 9, 203, 1, 1, 1, 1, 4, 23, 877, 1, 1, 1, 1, 2, 8, 65, 4140, 1, 1, 1, 1, 1, 4, 17, 199, 21147, 1, 1, 1, 1, 1, 2, 8, 40, 654, 115975, 1, 1, 1, 1, 1, 1, 4, 16, 104, 2296, 678570, 1, 1, 1, 1, 1, 1, 2, 8, 33, 291, 8569, 4213597, 1, 1, 1, 1

Offset: 1

R. H. Hardin, Mar 22 2012

			Some solutions for n=13 k=4
..0....0....0....0....0....0....0....0....0....0....0....0....0....0....0....0
..0....0....0....0....0....0....0....0....0....0....0....0....0....0....0....0
..0....0....0....0....0....0....0....0....0....0....0....0....0....0....0....0
..0....0....0....0....0....0....0....0....0....0....0....0....0....0....0....0
..1....1....1....1....1....1....0....0....1....0....0....1....1....0....1....1
..0....0....1....1....1....1....1....0....0....0....1....1....0....1....1....0
..1....0....0....1....1....1....0....1....1....0....0....0....0....1....1....0
..0....1....1....0....1....1....0....1....0....0....1....0....0....1....1....1
..1....1....0....0....1....2....1....1....0....1....1....0....0....1....2....1
..0....1....0....1....0....1....1....1....1....0....1....1....1....1....2....1
..0....1....0....2....1....0....0....1....1....0....2....0....1....1....2....1
..1....2....0....1....1....0....1....0....0....1....1....0....0....2....1....0
..0....2....1....0....2....0....0....0....2....1....2....1....0....0....0....1
Table starts
..........1.......1......1.....1....1...1...1
..........2.......1......1.....1....1...1...1
..........5.......2......1.....1....1...1...1
.........15.......4......2.....1....1...1...1
.........52.......9......4.....2....1...1...1
........203......23......8.....4....2...1...1
........877......65.....17.....8....4...2...1
.......4140.....199.....40....16....8...4...2
......21147.....654....104....33...16...8...4
.....115975....2296....291....73...32..16...8
.....678570....8569....857...177...65..32..16
....4213597...33825...2634...467..138..64..32
...27644437..140581...8455..1309..315.129..64
..190899322..612933..28424..3813..782.267.128
.1382958545.2795182.100117.11409.2090.582.257

R. H. Hardin, Table of n, a(n) for n = 1..9999
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.

Cf. A000110 (column 1), A007476 (column 2), A210540 (column 3).

T(n,k)=1 if n<=k else Sum_{i=0..n-k} binomial(n-k,i)*T(i,k). Proved by R. J. Mathar in the Sequence Fans Mailing List.

A245163 T(n,k)=Number of length n 0..k arrays with new values introduced in order from both ends.

Original entry on oeis.org

1, 1, 1, 1, 1, 2, 1, 1, 2, 4, 1, 1, 2, 4, 8, 1, 1, 2, 4, 9, 16, 1, 1, 2, 4, 9, 23, 32, 1, 1, 2, 4, 9, 23, 64, 64, 1, 1, 2, 4, 9, 23, 65, 186, 128, 1, 1, 2, 4, 9, 23, 65, 199, 551, 256, 1, 1, 2, 4, 9, 23, 65, 199, 653, 1645, 512, 1, 1, 2, 4, 9, 23, 65, 199, 654, 2275, 4926, 1024, 1, 1, 2, 4, 9, 23

Offset: 1

R. H. Hardin, Jul 12 2014

Table starts
.....1........1.........1.........1.........1.........1.........1.........1
.....1........1.........1.........1.........1.........1.........1.........1
.....2........2.........2.........2.........2.........2.........2.........2
.....4........4.........4.........4.........4.........4.........4.........4
.....8........9.........9.........9.........9.........9.........9.........9
....16.......23........23........23........23........23........23........23
....32.......64........65........65........65........65........65........65
....64......186.......199.......199.......199.......199.......199.......199
...128......551.......653.......654.......654.......654.......654.......654
...256.....1645......2275......2296......2296......2296......2296......2296
...512.....4926......8313......8568......8569......8569......8569......8569
..1024....14768.....31439.....33794.....33825.....33825.....33825.....33825
..2048....44293....121637....140039....140580....140581....140581....140581
..4096...132867....477307....605869....612890....612933....612933....612933
..8192...398588...1888721...2718531...2794159...2795181...2795182...2795182
.16384..1195750...7509799..12564289..13280627..13298407..13298464..13298464
.32768..3587235..29940861..59419764..65597882..65851100..65852872..65852873
.65536.10761689.119550419.285878342.335521900.338654554.338694406.338694479

			Some solutions for n=10 k=4
..0....0....0....0....0....0....0....0....0....0....0....0....0....0....0....0
..1....1....1....1....0....1....0....1....0....1....1....0....0....1....1....0
..1....0....2....0....1....2....1....0....0....2....2....0....1....1....2....1
..0....1....3....1....2....0....0....2....0....2....1....1....0....1....2....0
..1....2....0....0....0....1....2....3....0....3....1....2....1....1....2....1
..0....2....0....2....2....1....0....3....1....1....0....1....2....2....2....0
..0....1....2....1....2....1....0....2....1....3....2....1....1....1....1....2
..0....1....1....2....1....1....1....0....1....2....1....1....1....2....1....1
..1....1....1....1....1....1....0....1....0....1....1....0....1....1....1....1
..0....0....0....0....0....0....0....0....0....0....0....0....0....0....0....0

R. H. Hardin, Table of n, a(n) for n = 1..9999

Column 1 is A000079(n-2)
Column 2 is A164039(n-2)
Diagonal is A007476

Empirical for column k:
k=1: a(n) = 2*a(n-1) for n>2
k=2: a(n) = 5*a(n-1) -7*a(n-2) +3*a(n-3) for n>4
k=3: a(n) = 10*a(n-1) -37*a(n-2) +64*a(n-3) -52*a(n-4) +16*a(n-5) for n>6
k=4: [order 7] for n>8
k=5: [order 9] for n>10
k=6: [order 11] for n>12
k=7: [order 13] for n>14

A344489 a(n) = 1 + Sum_{k=0..n-2} binomial(n-1,k) * a(k).

Original entry on oeis.org

1, 1, 2, 4, 11, 34, 122, 487, 2144, 10276, 53165, 294760, 1740950, 10899841, 72033470, 500664496, 3648211139, 27792215302, 220802394110, 1825428024367, 15672798590804, 139499676115312, 1285109772354941, 12235037442987028, 120220980122266010, 1217655627762149857

Offset: 0

Ilya Gutkovskiy, May 21 2021

nonn

Cf. A000296, A000629, A007476, A186021, A337186, A344490, A344491, A344492, A344493.

a[n_] := a[n] = 1 + Sum[Binomial[n - 1, k] a[k] , {k, 0, n - 2}]; Table[a[n], {n, 0, 25}]
nmax = 25; A[] = 0; Do[A[x] = (1 + x A[x/(1 - x)])/(1 - x^2) + O[x]^(nmax + 1) // Normal, nmax + 1]; CoefficientList[A[x], x]

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

A351056 G.f. A(x) satisfies: A(x) = 1 + x + x^2 * A(x/(1 - 5x)) / (1 - 5x).

Original entry on oeis.org

1, 1, 1, 6, 36, 221, 1431, 10121, 80311, 718106, 7111976, 76201501, 868288401, 10438492181, 132166853861, 1763179150946, 24776241643056, 365971430085021, 5662954240306111, 91450179009971181, 1536249848608545451, 26782376261726525126, 483792982362049317676

Offset: 0

Ilya Gutkovskiy, Jan 30 2022

nonn

Shifts 2 places left under 5th-order binomial transform.

Cf. A005011, A007472, A007476, A351049, A351050, A351057.

nmax = 22; A[] = 0; Do[A[x] = 1 + x + x^2 A[x/(1 - 5 x)]/(1 - 5 x) + O[x]^(nmax + 1) // Normal, nmax + 1]; CoefficientList[A[x], x]
a[0] = a[1] = 1; a[n_] := a[n] = Sum[Binomial[n - 2, k] 5^k a[n - k - 2], {k, 0, n - 2}]; Table[a[n], {n, 0, 22}]

a(0) = a(1) = 1; a(n) = Sum_{k=0..n-2} binomial(n-2,k) * 5^k * a(n-k-2).

A351057 G.f. A(x) satisfies: A(x) = 1 + x + x^2 * A(x/(1 - 6x)) / (1 - 6x).

Original entry on oeis.org

1, 1, 1, 7, 49, 349, 2593, 20755, 184609, 1851289, 20735041, 253471039, 3310505425, 45630322741, 660993079393, 10065000586507, 161262522401089, 2717539655666353, 48053169836707969, 888408313419305719, 17108882037936283249, 342144175940842590349, 7089944927940141776545

Offset: 0

Ilya Gutkovskiy, Jan 30 2022

nonn

Shifts 2 places left under 6th-order binomial transform.

Cf. A005012, A007472, A007476, A351049, A351050, A351056.

nmax = 22; A[] = 0; Do[A[x] = 1 + x + x^2 A[x/(1 - 6 x)]/(1 - 6 x) + O[x]^(nmax + 1) // Normal, nmax + 1]; CoefficientList[A[x], x]
a[0] = a[1] = 1; a[n_] := a[n] = Sum[Binomial[n - 2, k] 6^k a[n - k - 2], {k, 0, n - 2}]; Table[a[n], {n, 0, 22}]

a(0) = a(1) = 1; a(n) = Sum_{k=0..n-2} binomial(n-2,k) * 6^k * a(n-k-2).

cp's OEIS Frontend

A153859 Triangle read by rows, A007318 * (A007476 * 0 ^(n-k)).

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A000994 Shifts 2 places left under binomial transform.

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A351049 G.f. A(x) satisfies: A(x) = 1 + x + x^2 * A(x/(1 - 3*x)) / (1 - 3*x).

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A351050 G.f. A(x) satisfies: A(x) = 1 + x + x^2 * A(x/(1 - 4*x)) / (1 - 4*x).

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A351437 G.f. A(x) satisfies: A(x) = 1 + x + x^2 * A(x/(1 - x)) / (1 - x)^2.

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A210545 T(n,k) = number of arrays of n nonnegative integers with value i>0 appearing only after i-1 has appeared at least k times.

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Formula

A245163 T(n,k)=Number of length n 0..k arrays with new values introduced in order from both ends.

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A344489 a(n) = 1 + Sum_{k=0..n-2} binomial(n-1,k) * a(k).

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A351056 G.f. A(x) satisfies: A(x) = 1 + x + x^2 * A(x/(1 - 5*x)) / (1 - 5*x).

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A351049 G.f. A(x) satisfies: A(x) = 1 + x + x^2 * A(x/(1 - 3x)) / (1 - 3x).

A351050 G.f. A(x) satisfies: A(x) = 1 + x + x^2 * A(x/(1 - 4x)) / (1 - 4x).

A351056 G.f. A(x) satisfies: A(x) = 1 + x + x^2 * A(x/(1 - 5x)) / (1 - 5x).

A351057 G.f. A(x) satisfies: A(x) = 1 + x + x^2 * A(x/(1 - 6x)) / (1 - 6x).