A103284 -id:A103284 - cp's OEIS frontend

A103285 Main diagonal of triangle A103284, in which row n+1 is formed by sorting, in ascending order, the result of the convolution of row n with {1,1}.

1, 1, 2, 3, 5, 9, 16, 29, 53, 97, 179, 333, 623, 1172, 2215, 4201, 7992, 15241, 29122, 55745, 106879, 205223, 394637, 759973, 1465589, 2830273, 5473068, 10597365, 20544860, 39876331, 77481582, 150700968, 293380151, 571619362, 1114581628

Offset: 0

Paul D. Hanna, Jan 28 2005

nonn

Paul D. Hanna, Table of n, a(n) for n = 0..1000

Cf. A103284.

a103285 = last . a103284_row  -- Reinhard Zumkeller, Nov 19 2015

{a(n)=local(A=vector(n+1,i,vector(i)),B);A[1][1]=1; for(k=1,n,B=vector(k+1);B[1]=1;B[k+1]=A[k][k]; for(i=2,k,B[i]=A[k][i]+A[k][i-1]); A[k+1]=vecsort(B));return(A[n+1][n+1])}

a(n) ~ 2^n*log_2(n)/(4n).

A257641 Central terms of triangle A103284.

Original entry on oeis.org

1, 1, 3, 9, 30, 102, 352, 1240, 4443, 16112, 58900, 216727, 801880, 2982327, 11147622, 41854916, 157805863, 597307573, 2268487476, 8639848112, 32999949539, 126498330700, 485441076004, 1865211479514, 7176111235446, 27644828367182, 106633724636570

Offset: 0

Reinhard Zumkeller, Nov 19 2015

nonn

a(n) = A103284(2*n,n).

Reinhard Zumkeller, Table of n, a(n) for n = 0..1000

Cf. A103284, A103285, A000984.

Haskell
```
a257641 n = a103284 (2 * n) n
```

A257646 Index of first row of triangle A103284 containing n.

Original entry on oeis.org

0, 2, 3, 4, 4, 5, 5, 9, 5, 11, 6, 13, 6, 7, 16, 6, 18, 7, 20, 7, 22, 23, 24, 7, 26, 27, 9, 29, 7, 8, 32, 33, 34, 8, 10, 37, 38, 8, 40, 41, 42, 43, 44, 8, 46, 47, 48, 49, 9, 51, 52, 53, 8, 12, 56, 57, 58, 59, 9, 61, 62, 63, 64, 9, 13, 67, 68, 69, 70, 71, 72

Offset: 1

Reinhard Zumkeller, Nov 19 2015

nonn

a(n) <= n + 1.

Reinhard Zumkeller, Table of n, a(n) for n = 1..1000

Cf. A103284, A058084.

import Data.List (findIndex); import Data.Maybe (fromJust)
a257646 n = fromJust $ findIndex (elem n) a103284_tabl

A107430 Triangle read by rows: row n is row n of Pascal's triangle (A007318) sorted into increasing order.

Original entry on oeis.org

1, 1, 1, 1, 1, 2, 1, 1, 3, 3, 1, 1, 4, 4, 6, 1, 1, 5, 5, 10, 10, 1, 1, 6, 6, 15, 15, 20, 1, 1, 7, 7, 21, 21, 35, 35, 1, 1, 8, 8, 28, 28, 56, 56, 70, 1, 1, 9, 9, 36, 36, 84, 84, 126, 126, 1, 1, 10, 10, 45, 45, 120, 120, 210, 210, 252, 1, 1, 11, 11, 55, 55, 165, 165, 330, 330, 462, 462, 1

Offset: 0

Philippe Deléham, May 21 2005

By rows, equals partial sums of A053121 reversed rows. Example: Row 4 of A053121 = (2, 0, 3, 0, 1) -> (1, 0, 3, 0, 2) -> (1, 1, 4, 4, 6). - Gary W. Adamson, Dec 28 2008, edited by Michel Marcus, Sep 22 2015

			Triangle begins:
1;
1,1;
1,1,2;
1,1,3,3;
1,1,4,4,6;

Reinhard Zumkeller, Rows n = 0..125 of triangle, flattened
Index entries for triangles and arrays related to Pascal's triangle

A061554 is similar but with rows sorted into decreasing order.
Cf. A034868.
Cf. A053121. - Gary W. Adamson, Dec 28 2008
Cf. A103284.

import Data.List (sort)
a107430 n k = a107430_tabl !! n !! k
a107430_row n = a107430_tabl !! n
a107430_tabl = map sort a007318_tabl
-- Reinhard Zumkeller, May 26 2013

/* As triangle */ [[Binomial(n,Floor(k/2)) : k in [0..n]]: n in [0.. 15]]; // Vincenzo Librandi, Sep 22 2015

for n from 0 to 10 do sort([seq(binomial(n,k),k=0..n)]) od; # yields sequence in triangular form. - Emeric Deutsch, May 28 2005

Flatten[ Table[ Sort[ Table[ Binomial[n, k], {k, 0, n}]], {n, 0, 12}]] (* Robert G. Wilson v, May 28 2005 *)

for(n=0,20, for(k=0,n, print1(binomial(n,floor(k/2)), ", "))) \\ G. C. Greubel, May 22 2017

T(n,k) = C(n,floor(k/2)). - Paul Barry, Dec 15 2006; corrected by Philippe Deléham, Mar 15 2007

Sum_{k=0..n} T(n,k)*x^(n-k) = A127363(n), A127362(n), A127361(n), A126869(n), A001405(n), A000079(n), A127358(n), A127359(n), A127360(n) for x=-4,-3,-2,-1,0,1,2,3,4 respectively. - Philippe Deléham, Mar 29 2007

More terms from Emeric Deutsch and Robert G. Wilson v, May 28 2005

A115323 Triangle, read by rows, where row n+1 is formed by sorting, in ascending order, the result of the convolution of row n (consisting of 2n+1 terms) with [1,1,1].

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 2, 2, 3, 1, 2, 3, 4, 5, 5, 7, 1, 3, 6, 7, 9, 12, 12, 14, 17, 1, 4, 10, 16, 17, 22, 28, 31, 33, 38, 43, 1, 5, 15, 30, 43, 43, 55, 67, 81, 81, 92, 102, 114, 1, 6, 21, 50, 88, 114, 116, 141, 165, 203, 216, 229, 254, 275, 308, 1, 7, 28, 77, 159, 252, 308, 318, 371, 422

Offset: 0

Paul D. Hanna, Jan 20 2006

			Convolution of [1,1,1] with row 3 gives:
[1,1,1]*[1,2,3,4,5,5,7] = [1,3,6,9,12,14,17,12,7];
when sorted yields row 4: [1,3,6,7,9,12,12,14,17].
Triangle begins:
1;
1,1,1;
1,1,2,2,3;
1,2,3,4,5,5,7;
1,3,6,7,9,12,12,14,17;
1,4,10,16,17,22,28,31,33,38,43;
1,5,15,30,43,43,55,67,81,81,92,102,114;
1,6,21,50,88,114,116,141,165,203,216,229,254,275,308;
1,7,28,77,159,252,308,318,371,422,509,583,584,648,699,758,837; ...

Cf. A103284 (variant), A115324 (central terms), A115325 (diagonal).

T(n,k)=local(V);if(2*n=0,T(n-1,i-2))+if(i>0,T(n-1,i-1))+T(n-1,i)); V=vecsort(V);V[k+1]))

A103286 Triangle, read by rows, where row n+1 is formed by sorting, in ascending order, the result of the convolution of row n with {2,1}.

Original entry on oeis.org

1, 1, 2, 2, 2, 5, 4, 5, 6, 12, 8, 12, 14, 17, 30, 16, 30, 32, 40, 48, 77, 32, 76, 77, 94, 112, 136, 202, 64, 184, 202, 230, 265, 318, 384, 540, 128, 432, 540, 588, 662, 760, 901, 1086, 1464, 256, 992, 1464, 1512, 1716, 1912, 2182, 2562, 3073, 4014, 512, 2240

Offset: 0

Paul D. Hanna, Jan 28 2005

Row sums are powers of 3. Main diagonal is A103287.

			Convolution of row 4 {4,5,6,12} with {2,1} = {8,14,17,30,12};
sort to obtain row 5: {8,12,14,17,30}.
Rows begin:
1,
1,2,
2,2,5,
4,5,6,12,
8,12,14,17,30,
16,30,32,40,48,77,
32,76,77,94,112,136,202,
64,184,202,230,265,318,384,540,
128,432,540,588,662,760,901,1086,1464,
256,992,1464,1512,1716,1912,2182,2562,3073,4014,...

Cf. A103287, A103284.

{T(n,k)=local(A=vector(n+1,i,vector(i)),B);A[1][1]=1; for(k=1,n,B=vector(k+1);B[1]=2*A[k][1];B[k+1]=A[k][k]; for(i=2,k,B[i]=2*A[k][i]+A[k][i-1]); A[k+1]=vecsort(B));return(A[n+1][k+1])}

A116908 Triangle, read by rows, where row n+1 is formed by sorting, in ascending order, the result of the convolution of row n with {1,2}.

Original entry on oeis.org

1, 1, 2, 1, 2, 3, 1, 3, 3, 5, 1, 4, 5, 6, 8, 1, 5, 8, 9, 11, 14, 1, 6, 13, 14, 17, 20, 24, 1, 7, 19, 24, 37, 31, 37, 44, 1, 8, 26, 43, 44, 51, 58, 68, 81, 1, 9, 34, 69, 81, 87, 95, 109, 126, 149, 1, 10, 43, 103, 149, 150, 168, 182, 204, 235, 274

Offset: 1

Jonathan Vos Post, Mar 16 2006

See also: A103284 Triangle, read by rows, where row n+1 is formed by sorting, in ascending order, the result of the convolution of row n with {1,1}. Main diagonal is: 1, 2, 3, 5, 8, 14, 24, 44, 81, 149, 274,... This is lexicographically second of an infinite sequence of triangles such as Paul D. Hanna's A103284.

			Convolution of row 5 {1,4,5,6,8} with {1,2} = {1,5,9,11,14,8}; sort to obtain row 6: {1,5,8,9,11,14}.
Rows begin:
1,
1,2,
1,2,3,
1,3,3,5,
1,4,5,6,8,
1,5,8,9,11,14,
1,6,13,14,17,20,24,
1,7,19,24,37,31,37,44,
1,8,26,43,44,51,58,68,81,
1,9,34,69,81,87,95,109,126,149,
1,10,43,103,149,150,168,182,204,235,274,...

Cf. A103284.

cp's OEIS Frontend

A103285 Main diagonal of triangle A103284, in which row n+1 is formed by sorting, in ascending order, the result of the convolution of row n with {1,1}.

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A257641 Central terms of triangle A103284.

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A257646 Index of first row of triangle A103284 containing n.

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A107430 Triangle read by rows: row n is row n of Pascal's triangle (A007318) sorted into increasing order.

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A115323 Triangle, read by rows, where row n+1 is formed by sorting, in ascending order, the result of the convolution of row n (consisting of 2n+1 terms) with [1,1,1].

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A103286 Triangle, read by rows, where row n+1 is formed by sorting, in ascending order, the result of the convolution of row n with {2,1}.

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PARI

A116908 Triangle, read by rows, where row n+1 is formed by sorting, in ascending order, the result of the convolution of row n with {1,2}.

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