A001919 -id:A001919 - cp's OEIS frontend

A071675 Array read by antidiagonals of trinomial coefficients.

1, 0, 1, 0, 1, 1, 0, 1, 2, 1, 0, 0, 3, 3, 1, 0, 0, 2, 6, 4, 1, 0, 0, 1, 7, 10, 5, 1, 0, 0, 0, 6, 16, 15, 6, 1, 0, 0, 0, 3, 19, 30, 21, 7, 1, 0, 0, 0, 1, 16, 45, 50, 28, 8, 1, 0, 0, 0, 0, 10, 51, 90, 77, 36, 9, 1, 0, 0, 0, 0, 4, 45, 126, 161, 112, 45, 10, 1, 0, 0, 0, 0, 1, 30, 141, 266, 266

Offset: 0

Henry Bottomley, May 30 2002

Read as a number triangle, this is the Riordan array (1, x(1+x+x^2)) with T(n,k) = Sum_{i=0..floor((n+k)/2)} C(k,2i+2k-n)*C(2i+2k-n,i). Rows start {1}, {0,1}, {0,1,1}, {0,1,2,1}, {0,0,3,3,1},... Row sums are then the trinomial numbers A000073(n+2). Diagonal sums are A013979. - Paul Barry, Feb 15 2005

Let {a_(k,i)}, k>=1, i=0,...,k, be the k-th antidiagonal of the array. Then s_k(n)=sum{i=0,...,k}a_(k,i)* binomial(n,k) is the n-th element of the k-th column of A213742. For example, s_1(n)=binomial(n,1)=n is the first column of A213742 for n>1, s_2(n)=binomial(n,1)+binomial(n,2)is the second column of A213742 for n>1, etc. In particular (see comment in A213742) in cases k=4,5,6,7,8, s_k(n) is A005718(n+2), A005719(n), A005720(n), A001919(n), A064055(n+3), respectively. - Vladimir Shevelev and Peter J. C. Moses, Jun 22 2012

			Rows start
1, 0,  0,  0,  0,  0, ...;
1, 1,  1,  0,  0,  0,  0, ...;
1, 2,  3,  2,  1,  0,  0, ...;
1, 3,  6,  7,  6,  3,  1, 0, ...;
1, 4, 10, 16, 19, 16, 10, 4, 1, ...; etc.

G. C. Greubel, Table of n, a(n) for the first 50 rows, flattened
Milan Janjic, Binomial Coefficients and Enumeration of Restricted Words, Journal of Integer Sequences, 2016, Vol 19, #16.7.3

Visible version of A027907. Row sums are 3^n, i.e. A000244. Central diagonal is A002426. Cf. A071676 for a slight variation.

T[n_, k_] := Sum[Binomial[n - k - j, j]*Binomial[k, n - k - j], {j, 0,
Floor[(n - k)/2]}]; Table[T[n, k], {n,0,10}, {k,0,n}] // Flatten (* G. C. Greubel, Feb 28 2017 *)

T(n, k) = T(n-1, k) + T(n-1, k-1) + T(n-1, k-2) starting with T(0, 0)=1. See A027907 for more.

As a number triangle, T(n, k) = Sum_{i=0..floor((n-k)/2)} C(n-k-i, i) * C(k, n-k-i). - Paul Barry, Apr 26 2005

A213742 Triangle of numbers C^(3)(n,k) of combinations with repetitions from n different elements over k for each of them not more than three appearances allowed.

Original entry on oeis.org

1, 1, 1, 1, 2, 3, 1, 3, 6, 10, 1, 4, 10, 20, 31, 1, 5, 15, 35, 65, 101, 1, 6, 21, 56, 120, 216, 336, 1, 7, 28, 84, 203, 413, 728, 1128, 1, 8, 36, 120, 322, 728, 1428, 2472, 3823, 1, 9, 45, 165, 486, 1206, 2598, 4950, 8451, 13051, 1, 10

Offset: 0

Vladimir Shevelev and Peter J. C. Moses, Jun 19 2012

The left side of triangle consists of 1's, while the right side is formed by A005725. Further, T(n,0)=1, T(n,1)=n, T(n,2)=A000217(n) for n>1, T(n,3)=A000292(n) for n>=3, T(n,4)=A005718(n) for n>=2, T(n,5)=A005719(n) for n>=5, T(n,6)=A005720(n) for n>=6, T(n,7)=A001919(n) for n>=7, T(n,8)=A064055(n) for n>=5.

			Triangle begins
n/k.|..0.....1.....2.....3.....4.....5.....6.....7
==================================================
.0..|..1
.1..|..1.....1
.2..|..1.....2.....3
.3..|..1.....3.....6....10
.4..|..1.....4....10....20....31
.5..|..1.....5....15....35....65....101
.6..|..1.....6....21....56...120....216...336
.7..|..1.....7....28....84...203....413...728....1128

Peter J. C. Moses, Rows n = 0..50 of triangle, flattened

Cf. A007318, A005725, A111808, A000217, A000292, A005718, A005719, A005720, A001919, A064055.

Flatten[Table[Sum[(-1)^r Binomial[n,r] Binomial[n-# r+k-1,n-1],{r,0,Floor[k/#]}],{n,0,15},{k,0,n}]/.{0}->{1}]&[4] (* Peter J. C. Moses, Apr 16 2013 *)

C^(3)(n,k)=sum{r=0,...,floor(k/4)}(-1)^r*C(n,r)*C(n-4*r+k-1, n-1)

A278734 T(n,k)=Number of nXk 0..3 arrays with rows in nondecreasing lexicographic order and columns in nonincreasing lexicographic order, but with exactly one mistake.

Original entry on oeis.org

0, 6, 6, 40, 152, 40, 155, 1947, 1947, 155, 456, 17352, 58904, 17352, 456, 1128, 121520, 1410818, 1410818, 121520, 1128, 2472, 712406, 28637916, 99992428, 28637916, 712406, 2472, 4950, 3633649, 506031118, 6410559865, 6410559865, 506031118

Offset: 1

R. H. Hardin, Nov 27 2016

Table starts
.....0.........6.............40.................155......................456
.....6.......152...........1947...............17352...................121520
....40......1947..........58904.............1410818.................28637916
...155.....17352........1410818............99992428...............6410559865
...456....121520.......28637916..........6410559865............1351385130108
..1128....712406......506031118........374757577056..........268284486351027
..2472...3633649.....7907770636......19983433877142........50067074390669892
..4950..16547278...110655824716.....971720519011047......8732216738504713198
..9240..68531079..1401584381570...43159978267689118...1415177080112634284232
.16302.261693631.16222274394016.1757375854436887414.212485358907612452321760

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

R. H. Hardin, Table of n, a(n) for n = 1..111

Column 1 is A001919(n+1).

Empirical for column k:
k=1: [polynomial of degree 7]
k=2: [polynomial of degree 28]
k=3: [polynomial of degree 109]

A278435 T(n,k)=Number of nXk 0..3 arrays with rows and columns in lexicographic nondecreasing order but with exactly one mistake.

Original entry on oeis.org

0, 6, 6, 40, 100, 40, 155, 1609, 1609, 155, 456, 19624, 57760, 19624, 456, 1128, 178352, 2116789, 2116789, 178352, 1128, 2472, 1287838, 67971132, 223202074, 67971132, 1287838, 2472, 4950, 7795151, 1796061464, 23450120081, 23450120081, 1796061464

Offset: 1

R. H. Hardin, Nov 22 2016

Table starts
....0.......6.........40...........155..............456................1128
....6.....100.......1609.........19624...........178352.............1287838
...40....1609......57760.......2116789.........67971132..........1796061464
..155...19624....2116789.....223202074......23450120081.......2266913897519
..456..178352...67971132...23450120081....7817299555828....2573951428892959
.1128.1287838.1796061464.2266913897519.2573951428892959.2817080307689646420

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

R. H. Hardin, Table of n, a(n) for n = 1..97

Column 1 is A001919(n+1).

Empirical for column k:
k=1: [polynomial of degree 7]
k=2: [polynomial of degree 31]
k=3: [polynomial of degree 127]

A064055 Ninth column of quadrinomial coefficients.

Original entry on oeis.org

3, 31, 155, 546, 1554, 3823, 8451, 17205, 32802, 59268, 102388, 170261, 273975, 428418, 653242, 973998, 1423461, 2043165, 2885169, 4014076, 5509328, 7467801, 10006725, 13266955, 17416620, 22655178, 29217906

Offset: 0

Wolfdieter Lang, Aug 29 2001

A001919 (eighth column).

Table[3Binomial[n+3,3]+19Binomial[n+3,4]+30Binomial[n+3,5]+21 Binomial[n+3,6]+ 7 Binomial[n+3,7]+ Binomial[n+3,8],{n,0,30}] (* Harvey P. Dale, Apr 30 2022 *)

a(n)= A008287(n+3, 8)= binomial(n+3, 3)*(n^5+46*n^4+875*n^3+7118*n^2+23880*n+20160)/(8!/3!), n >= 0.
G.f.: (3+4*x-16*x^2+15*x^3-6*x^4+x^5 )/(1-x)^9, numerator polynomial is N4(8, x) from the array A063421.
a(n) = 3*C(n+3,3) + 19*C(n+3,4) + 30*C(n+3,5) + 21*C(n+3,6) + 7*C(n+3,7) + C(n+3,8) (see comment in A071675). - Vladimir Shevelev and Peter J. C. Moses, Jun 22 2012

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A071675 Array read by antidiagonals of trinomial coefficients.

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A213742 Triangle of numbers C^(3)(n,k) of combinations with repetitions from n different elements over k for each of them not more than three appearances allowed.

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A278734 T(n,k)=Number of nXk 0..3 arrays with rows in nondecreasing lexicographic order and columns in nonincreasing lexicographic order, but with exactly one mistake.

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A278435 T(n,k)=Number of nXk 0..3 arrays with rows and columns in lexicographic nondecreasing order but with exactly one mistake.

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Comments

Examples

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Crossrefs

Formula

A064055 Ninth column of quadrinomial coefficients.

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