A051925 -id:A051925 - cp's OEIS frontend

A208519 Triangle of coefficients of polynomials v(n,x) jointly generated with A208518; see the Formula section.

1, 2, 2, 3, 5, 3, 4, 9, 11, 5, 5, 14, 26, 23, 8, 6, 20, 50, 65, 45, 13, 7, 27, 85, 145, 150, 86, 21, 8, 35, 133, 280, 385, 329, 160, 34, 9, 44, 196, 490, 840, 952, 692, 293, 55, 10, 54, 276, 798, 1638, 2310, 2232, 1413, 529, 89, 11, 65, 375, 1230, 2940, 4956

Offset: 1

Clark Kimberling, Feb 28 2012

coefficient of x^(n-1): Fibonacci(n+1) = A000045(n+1)
col 1:  A000027
col 2:  A000096
col 3:  A051925
row sums:  A002878 (bisection of Lucas sequence)
alternating row sums:  A000045(n-2), Fibonacci numbers

			First five rows:
1
2...2
3...5....3
4...9....11...5
5...14...26...23...8
First five polynomials v(n,x):
1
2 + 2x
3 + 5x + 3x^2
4 + 9x + 11x^2 + 5x^3
5 + 14x + 26x^2 + 23x^3 + 8x^4

Cf. A208518.

u[1, x_] := 1; v[1, x_] := 1; z = 16;
u[n_, x_] := u[n - 1, x] + x*v[n - 1, x];
v[n_, x_] := x*u[n - 1, x] + (x + 1)*v[n - 1, x] + 1;
Table[Expand[u[n, x]], {n, 1, z/2}]
Table[Expand[v[n, x]], {n, 1, z/2}]
cu = Table[CoefficientList[u[n, x], x], {n, 1, z}];
TableForm[cu]
Flatten[%]   (* A208518 *)
Table[Expand[v[n, x]], {n, 1, z}]
cv = Table[CoefficientList[v[n, x], x], {n, 1, z}];
TableForm[cv]
Flatten[%]   (* A208519 *)

u(n,x)=u(n-1,x)+x*v(n-1,x),
v(n,x)=x*u(n-1,x)+(x+1)*v(n-1,x)+1,
where u(1,x)=1, v(1,x)=1.

A210755 Triangle of coefficients of polynomials u(n,x) jointly generated with A210756; see the Formula section.

Original entry on oeis.org

1, 2, 3, 3, 11, 7, 4, 26, 38, 17, 5, 50, 124, 121, 41, 6, 85, 314, 499, 362, 99, 7, 133, 679, 1555, 1805, 1043, 239, 8, 196, 1316, 4054, 6672, 6096, 2926, 577, 9, 276, 2352, 9318, 20326, 26048, 19610, 8049, 1393, 10, 375, 3948, 19482, 53932, 90706

Offset: 1

Clark Kimberling, Mar 25 2012

Row n starts with n and ends with A001333(n).
Column 2: A051925
Row sums: A002450
Alternating row sums: 1,-1,-1,-1,-1,-1,...
For a discussion and guide to related arrays, see A208510.

			First five rows:
1
2...3
3...11...7
4...26...38....17
5...50...124...121...41
First three polynomials u(n,x): 1, 2 + 3x, 3 + 11x + 7x^2.

Cf. A210754, A208510.

u[1, x_] := 1; v[1, x_] := 1; z = 16;
u[n_, x_] := (x + 1)*u[n - 1, x] + 2 x*v[n - 1, x] + 1;
v[n_, x_] := (x + 1)*u[n - 1, x] + (x + 1)*v[n - 1, x] + 1;
Table[Expand[u[n, x]], {n, 1, z/2}]
Table[Expand[v[n, x]], {n, 1, z/2}]
cu = Table[CoefficientList[u[n, x], x], {n, 1, z}];
TableForm[cu]
Flatten[%]    (* A210755 *)
Table[Expand[v[n, x]], {n, 1, z}]
cv = Table[CoefficientList[v[n, x], x], {n, 1, z}];
TableForm[cv]
Flatten[%]    (* A210756 *)
Table[u[n, x] /. x -> 1, {n, 1, z}]  (* A002450 *)
Table[v[n, x] /. x -> 1, {n, 1, z}]  (* A002450 *)

u(n,x)=(x+1)*u(n-1,x)+2x*v(n-1,x)+1,
v(n,x)=(x+1)*u(n-1,x)+(x+1)*v(n-1,x)+1,
where u(1,x)=1, v(1,x)=1.

A104634 Triangle T(n,k) = (k-1-n)(k-2-n)(k+2*n)/6, 1<=k<=n.

Original entry on oeis.org

1, 5, 2, 14, 8, 3, 30, 20, 11, 4, 55, 40, 26, 14, 5, 91, 70, 50, 32, 17, 6, 140, 112, 85, 60, 38, 20, 7, 204, 168, 133, 100, 70, 44, 23, 8, 285, 240, 196, 154, 115, 80, 50, 26, 9, 385, 330, 276, 224, 175, 130, 90, 56, 29, 10, 506, 440, 375, 312, 252, 196, 145, 100, 62, 32, 11, 650, 572, 495, 420, 348, 280, 217, 160, 110, 68, 35, 12, 819, 728, 638, 550, 465, 384

Offset: 1

Gary W. Adamson, Mar 18 2005

			The first few rows are:
1;
5, 2;
14, 8, 3;
30, 20, 11, 4;
55, 40, 26, 14, 5;
91, 70, 50, 32, 17, 6;
...

G. C. Greubel, Rows n=1..100 of triangle, flattened

Cf. A000330 (column 1), A007290 (column 2), A051925 (column 3), A001296 (row sums), A104633, A000332.

[[(k-1-n)*(k-2-n)*(k+2*n)/6: k in [1..n]]: n in [1..20]]; // G. C. Greubel, Aug 12 2018

A104634 := proc(n,k) (k-1-n)*(k-2-n)*(k+2*n)/6 ; end proc:
seq(seq(A104634(n,k),k=1..n),n=1..15) ; # R. J. Mathar, Aug 31 2011

Table[(k-1-n)*(k-2-n)*(k+2*n)/6, {n, 1, 20}, {k, 1, n}] // Flatten (* G. C. Greubel, Aug 12 2018 *)

for(n=1,20, for(k=1,n, print1((k-1-n)*(k-2-n)*(k+2*n)/6, ", "))) \\ G. C. Greubel, Aug 12 2018

The triangle is created by the matrix product A002260 * A004736, both infinite lower triangular matrices.

Definition in closed form provided by R. J. Mathar, Aug 31 2011

A110952 Triangle read by rows: T(n,k) = number of permutations of [n] where the first increasing run has length k and the last increasing run has length n-k-1, 0

Original entry on oeis.org

1, 3, 3, 6, 11, 6, 10, 26, 26, 10, 15, 50, 71, 50, 15, 21, 85, 155, 155, 85, 21, 28, 133, 295, 379, 295, 133, 28, 36, 196, 511, 799, 799, 511, 196, 36, 45, 276, 826, 1519, 1849, 1519, 826, 276, 45, 55, 375, 1266, 2674, 3829, 3829, 2674, 1266, 375, 55, 66, 495, 1860

Offset: 3

David Scambler, Nov 22 2006

Permutations of [n] with exactly 2 descents and the descents are adjacent. Adjusting for initial index: row sums are A045618; first diagonal is A000217, the triangular numbers; 2nd diagonal is A051925; and 3rd diagonal is A001701, generalized Stirling numbers.

			Triangle (beginning with n=3, k=1) is:
   1
   3  3
   6 11  6
  10 26 26 10
  15 50 71 50 15
  ...
For n=5, k = 2: T(5,2) = 11 = permutations of [5] with first run 2 long and last run 5-2-1 = 2 long, namely {14325, 15324, 15423, 24315, 25314, 25413, 34215, 35214, 35412, 45213, 45312}.

Joseph Adams, Table of n, a(n) for n = 3..4755

Cf. A045618, A000217, A051925, A001701, A112858.

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

A264750 Number of sequences of 5 throws of an n-sided die (with faces numbered 1, 2, ..., n) in which the sum of the throws first reaches or exceeds n on the 5th throw.

Original entry on oeis.org

5, 29, 99, 259, 574, 1134, 2058, 3498, 5643, 8723, 13013, 18837, 26572, 36652, 49572, 65892, 86241, 111321, 141911, 178871, 223146, 275770, 337870, 410670, 495495, 593775, 707049, 836969, 985304, 1153944, 1344904, 1560328, 1802493, 2073813, 2376843, 2714283

Offset: 5

Louis Rogliano, Nov 23 2015

Sequence gives the second column of A185508. [Bruno Berselli, Nov 24 2015]

Number of 5-tuples (t_1, ..., t_5) with 1 <= t_j <= n, Sum_{j <= 4} t_j < n and Sum_{j<=5} t_j >= n. - Robert Israel, Nov 25 2015

			From _Jon E. Schoenfield_, Nov 26 2015: (Start)
For n=5, the a(5) = 5 sequences (i.e., quintuples or 5-tuples) are {1,1,1,1,1}, {1,1,1,1,2}, {1,1,1,1,3}, {1,1,1,1,4} and {1,1,1,1,5}. (Each of the first four throws must be a 1; otherwise, the sum of the throws would reach or exceed 5 before the 5th throw.)
For n=6, each of the quintuples must have four throws whose sum is less than 6, followed by a fifth throw that brings the sum to at least 6, so the a(6) = 29 quintuples are the 5 quintuples {1,1,1,1,t_5} where t_5 is any value in 2..6 and the four sets of 6 quintuples {1,1,1,2,t_5}, {1,1,2,1,t_5}, {1,2,1,1,t_5} and {2,1,1,1,t_5} where t_5 is any value in 1..6. (End)

Colin Barker, Table of n, a(n) for n = 5..1000
Louis Rogliano, Sequence A264750
Index entries for linear recurrences with constant coefficients, signature (6,-15,20,-15,6,-1).

Cf. A000096 (k=2), A051925 (k=3), A215862 (k=4).

Cf. A185508.

[(n-4)*(n-3)*(n-2)*(n-1)*(4*n+5)/120: n in [5..40]]; // Vincenzo Librandi, Nov 24 2015

A264750:=n->(n-4)*(n-3)*(n-2)*(n-1)*(4*n+5)/120: seq(A264750(n), n=5..50); # Wesley Ivan Hurt, Nov 24 2015

f[n_, k_] := Module[
{i, total = 0, part, perm},
part = IntegerPartitions[n, {k}];
perm = Flatten[Table[Permutations[part[[i]]], {i, 1, Length[part]}],      1];
For[i = 1, i <= Length[perm], i++,    total += n + 1 - perm[[i, k]]    ];
Return[total];   ]
And the sequences are obtained by:
h[k_] := Table[f[i, k], {i, k, number_of_terms_wanted}]
Table[(n - 4) (n - 3) (n - 2) (n - 1) (4 n + 5)/120, {n, 5, 40}] (* Bruno Berselli, Nov 24 2015 *)

Vec(x^5*(5-x)/(1-x)^6 + O(x^100)) \\ Colin Barker, Nov 23 2015

for(n=5, 40, print1((n-4)*(n-3)*(n-2)*(n-1)*(4*n+5)/120", ")); \\ Bruno Berselli, Nov 24 2015

[(n-4)*(n-3)*(n-2)*(n-1)*(4*n+5)/120 for n in (5..40)] # Bruno Berselli, Nov 24 2015

From Colin Barker, Nov 23 2015: (Start)
a(n) = (n - 4)*(n - 3)*(n - 2)*(n - 1)*(4*n + 5)/120.
a(n) = 6*a(n-1) - 15*a(n-2) + 20*a(n-3) - 15*a(n-4) + 6*a(n-5) - a(n-6) for n>10.
G.f.: x^5*(5 - x) / (1 - x)^6. (End)

Offset changed by Robert Israel, Nov 25 2015

Formulae, b-file adapted to the new offset and definition rephrased by the Editors of the OEIS, Nov 26 2015

A047732 First differences are A005563.

Original entry on oeis.org

1, 4, 12, 27, 51, 86, 134, 197, 277, 376, 496, 639, 807, 1002, 1226, 1481, 1769, 2092, 2452, 2851, 3291, 3774, 4302, 4877, 5501, 6176, 6904, 7687, 8527, 9426, 10386, 11409, 12497, 13652, 14876, 16171, 17539, 18982, 20502, 22101, 23781, 25544, 27392, 29327

Offset: 0

Patternfinder(AT)webtv.net (Robert Newstedt)

Number of 3-permutations of n elements avoiding the patterns 132, 321. See Bonichon and Sun. - Michel Marcus, Aug 19 2022

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Nicolas Bonichon and Pierre-Jean Morel, Baxter d-permutations and other pattern avoiding classes, arXiv:2202.12677 [math.CO], 2022.
Nathan Sun, On d-permutations and Pattern Avoidance Classes, arXiv:2208.08506 [math.CO], 2022.
Index entries for linear recurrences with constant coefficients, signature (4,-6,4,-1).

Cf. A000330, A005563, A051925.

I:=[1, 4, 12, 27]; [n le 4 select I[n] else 4*Self(n-1)-6*Self(n-2)+4*Self(n-3)-Self(n-4): n in [1..40]]; // Vincenzo Librandi, Jun 28 2012

CoefficientList[Series[(1+2*x^2-x^3)/((1-x)^4),{x,0,50}],x] (* Vincenzo Librandi, Jun 28 2012 *)
LinearRecurrence[{4,-6,4,-1},{1,4,12,27},50] (* Harvey P. Dale, Aug 22 2015 *)

a(n) = A051925(n+1) + 1. - Alex Ratushnyak, Jun 27 2012
From Vincenzo Librandi, Jun 28 2012: (Start)
G.f.: (1 + 2*x^2 - x^3)/(1-x)^4.
a(n) = 4*a(n-1) - 6*a(n-2) + 4*a(n-3) - a(n-4).
a(n) = (2*n^3 + 9*n^2 + 7*n + 6)/6. (End)
a(n) = A000330(n+1) - n. - John Tyler Rascoe, Jun 24 2022

A145068 Zero followed by partial sums of A059100, starting at n=1.

Original entry on oeis.org

0, 3, 9, 20, 38, 65, 103, 154, 220, 303, 405, 528, 674, 845, 1043, 1270, 1528, 1819, 2145, 2508, 2910, 3353, 3839, 4370, 4948, 5575, 6253, 6984, 7770, 8613, 9515, 10478, 11504, 12595, 13753, 14980, 16278, 17649, 19095, 20618, 22220, 23903, 25669

Offset: 1

Vladimir Joseph Stephan Orlovsky, Sep 30 2008

			a(2) = a(1) + 1^2 + 2 = 0 + 1 + 2 = 3; a(3) = a(2) + 2^2 + 2 = 3 + 4 + 2 = 9.

Cf. A059100 (n^2+2), A002522 (n^2 + 1), A145066 (partial sums of A002522, starting at n=1), A008865 (n^2 - 2), A145067 (zero followed by partial sums of A008865), A005563 ((n+1)^2 - 1), A051925 (zero followed by partial sums of A005563).

lst={0};s=0;Do[s+=n^2+2;AppendTo[lst,s],{n,5!}];lst

{a=-2; for(n=0, 42, print1(a=a+n^2+2, ","))}

G.f.: x*(3-3*x+2*x^2)/(1-x)^4.

a(1) = 0; a(n) = sum_{j=1..n-1} A059100(j) = a(n-1) + (n-1)^2 + 2 for n > 1.

Edited by Klaus Brockhaus, Oct 21 2008

A194130 a(n) = n!/gcd(n,3).

Original entry on oeis.org

1, 2, 2, 24, 120, 240, 5040, 40320, 120960, 3628800, 39916800, 159667200, 6227020800, 87178291200, 435891456000, 20922789888000, 355687428096000, 2134124568576000, 121645100408832000, 2432902008176640000, 17030314057236480000, 1124000727777607680000

Offset: 1

Paul Curtz, Aug 16 2011

nonn

Cf. A109007, A144396, A005563, A051925.

A194130 := proc(n)
        n!/igcd(n,3) ;
end proc:
seq(A194130(n),n=1..30) ;

Table[n!/GCD[n,3],{n,20}] (* Harvey P. Dale, Aug 08 2019 *)

Definition and offset corrected by R. J. Mathar, Aug 18 2011

A206492 Sums of rows of the sequence of triangles with nonnegative integers and row widths defined by A004738.

Original entry on oeis.org

0, 3, 3, 9, 21, 19, 11, 25, 45, 74, 66, 49, 26, 55, 90, 134, 190, 170, 138, 97, 50, 103, 162, 230, 310, 405, 365, 310, 243, 167, 85, 173, 267, 370, 485, 615, 763, 693, 605, 502, 387, 263, 133, 269, 411, 562, 725, 903, 1099, 1316, 1204, 1071, 920, 754, 576, 389

Offset: 1

Alex Ratushnyak, Jun 28 2012

Row widths: A004738(n): 1, 2, 1, 2, 3, 2, 1, 2, 3, 4, 3, 2, 1, 2, 3, 4, 5...
Pits: A051925(n+1): 0, 3, 11, 26, 50, 85, 133, 196, 276, 375, 495, 638...
Peak tops: A007290(n+3): 2, 8, 20, 40, 70, 112, 168, 240, 330, 440, 572...
Peak bases: A084990(n+1): 1, 6, 17, 36, 65, 106, 161, 232, 321, 430, 561...

			The sequence of triangles begins:
0
1 2
3
4 5
6 7 8
9 10
11
12 13
14 15 16
17 18 19 20
21 22 23
24 25
26
27 28
29 30 31
32 33 34 35
36 37 38 39 40
41 42 43 44
45 46 47
48 49
50
51 52

Cf. A004738, A051925, A007290, A084990.

Cf. A027480: sums of rows of a triangle with increasing row widths: 0; 1,2; 3,4,5; 6,7,8,9; ...

curSign=-1
curLength=sum=0
rowLength=topLength=1
for n in range(1232):
    sum += n
    curLength += 1
    if curLength==rowLength:
        print(sum, end=',')
        curLength = sum = 0
        if rowLength==1 or rowLength==topLength:
            curSign = -curSign
        topLength += (rowLength==1)
        rowLength += curSign

A229834 Expansion of (1+4x+x^2) / ((1-x)^3(1+x)^4).

Original entry on oeis.org

1, 3, 1, 11, -2, 26, -10, 50, -25, 85, -49, 133, -84, 196, -132, 276, -195, 375, -275, 495, -374, 638, -494, 806, -637, 1001, -805, 1225, -1000, 1480, -1224, 1768, -1479, 2091, -1767, 2451, -2090, 2850, -2450, 3290, -2849, 3773, -3289, 4301, -3772, 4876, -4300, 5500, -4875, 6175, -5499, 6903, -6174, 7686, -6902

Offset: 0

Stefano Maruelli, Dec 19 2013

The sequence can be generated in the following way:
---------------------------       --------------------------
      [0]  [1] [2] [3]  [4]  ...             [i]
---------------------------       --------------------------
[0]    1,   1,  1,  1,   1,  ...  t(0,i) =    1
[1]    7,   6,  5,  4,   3,  ...  t(1,i) = t(1,i-1) - t(0,i)
[2]   19,  13,  8,  4,   1,  ...  t(2,i) = t(2,i-1) - t(1,i)
[3]   37,  24, 16, 12,  11,  ...  t(3,i) = t(3,i-1) - t(2,i)
[4]   61,  37, 21,  9,  -2,  ...  t(4,i) = t(4,i-1) - t(3,i)
[5]   91,  54, 33, 24,  26,  ...             etc.
[6]  127,  73, 40, 16, -10,  ...
[7]  169,  96, 56, 40,  50,  ...
[8]  217, 121, 65, 25, -25,  ...
[9]  271, 150, 85, 60,  85,  ...
...
Column 0 is A003215;
column 1 is A032528;
column 2 is A001082;
column 3 is A241496;
column 4 is this sequence.
The third differences are 16, -35, 64, -105, 160, ..., a signed variant of A077415. - R. J. Mathar, Apr 18 2014

Index entries for linear recurrences with constant coefficients, signature (-1,3,3,-3,-3,1,1).

Cf. A077415; A058373: a(2k) = -A058373(k); A051925: a(2k+1) = A051925(k+2).

Columns of the table in Comments section: A001082, A003215, A032528.

Table[1 + n (n + 5) (9 - (2 n + 5) (-1)^n)/48, {n, 0, 60}] (* Bruno Berselli, Apr 22 2014 *)
CoefficientList[Series[(1+4x+x^2)/((1-x)^3(1+x)^4),{x,0,60}],x] (* or *) LinearRecurrence[{-1,3,3,-3,-3,1,1},{1,3,1,11,-2,26,-10},60] (* Harvey P. Dale, Jan 27 2022 *)

G.f.: (1 + 4*x + x^2) / ((1 - x)^3*(1 + x)^4). - R. J. Mathar, Apr 18 2014

a(n) = a(-n-5) = 1 + n*(n + 5)*(9 - (2*n + 5)*(-1)^n)/48. [Bruno Berselli, Apr 22 2014]

cp's OEIS Frontend

A208519 Triangle of coefficients of polynomials v(n,x) jointly generated with A208518; see the Formula section.

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A210755 Triangle of coefficients of polynomials u(n,x) jointly generated with A210756; see the Formula section.

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A104634 Triangle T(n,k) = (k-1-n)*(k-2-n)*(k+2*n)/6, 1<=k<=n.

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A110952 Triangle read by rows: T(n,k) = number of permutations of [n] where the first increasing run has length k and the last increasing run has length n-k-1, 0

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A264750 Number of sequences of 5 throws of an n-sided die (with faces numbered 1, 2, ..., n) in which the sum of the throws first reaches or exceeds n on the 5th throw.

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A047732 First differences are A005563.

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A145068 Zero followed by partial sums of A059100, starting at n=1.

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A104634 Triangle T(n,k) = (k-1-n)(k-2-n)(k+2*n)/6, 1<=k<=n.

A229834 Expansion of (1+4x+x^2) / ((1-x)^3(1+x)^4).