A006011 -id:A006011 - cp's OEIS frontend

A159572 A triangular array related to ordered partitions and having row sums 1,2,5,14,43,144,523,2048,8597... A047970.

1, 1, 1, 1, 3, 1, 1, 6, 5, 2, 1, 10, 14, 12, 6, 1, 15, 30, 39, 39, 20, 1, 21, 55, 95, 138, 142, 71, 1, 28, 91, 195, 364, 548, 551, 270, 1, 36, 140, 357, 804, 1564, 2317, 2278, 1100

Offset: 1

Alford Arnold, Apr 16 2009

The first five diagonals are essentially A000012, A000217, A000330, A086602 and A159571.
From Alford Arnold, Apr 20 2009: (Start)
After the first two diagonals, each additional diagonal is computed using blocks of source partitions (defined in A053445).
The size of each block increases by powers of two; e.g. 22, 33 222, 44 332 333 2222; etc.
Each source partition can be associated with a specific sequence as illustrated in the below example using partition 332: grow the leftmost value to form 432 then append "1" to form 3321. in like manner, generate 532 4321 and 33211 from the previously formed cases. Note that the number of arrangements are 3, 6+12, and 6+24+30 respectively and that we now have three terms of A006011: 3 18 and 60.
Next we note that 6 39 138 364 804 ... A159571 resulted from summing term by term, the sequences associated with partitions 44 332 333 and 2222:
1...5..14...30...55
3..18..60..150..315
1...7..25...65..140
1...9..39..119..294
(End)

Cf. A047970 (row sums), A000012, A000217, A000330, A086602, A159571.

Cf. A053445. - Alford Arnold, Apr 20 2009

Submitted without a definition. - N. J. A. Sloane, Apr 18 2009

More terms from Alford Arnold, Oct 06 2009

A305402 A number triangle T(n,k) read by rows for 0<=k<=n, related to the Taylor expansion of f(u, p) = (1/2)(1+1/(sqrt(1-u^2)))exp(p*sqrt(1-u^2)).

Original entry on oeis.org

1, 1, -2, 3, -4, 2, 15, -18, 9, -2, 105, -120, 60, -16, 2, 945, -1050, 525, -150, 25, -2, 10395, -11340, 5670, -1680, 315, -36, 2, 135135, -145530, 72765, -22050, 4410, -588, 49, -2, 2027025, -2162160, 1081080, -332640, 69300, -10080, 1008, -64, 2

Offset: 0

Johannes W. Meijer, May 31 2018

The function f(u, p) = (1/2)*(1+1/(sqrt(1-u^2))) * exp(p*sqrt(1-u^2)) was found while studying the Fresnel-Kirchhoff and the Rayleigh-Sommerfeld theories of diffraction, see the Meijer link.
The Taylor expansion of f(u, p) leads to the number triangle T(n, k), see the example section.
Normalization of the triangle terms, dividing the T(n, k) by T(n-k, 0), leads to A084534.
The row sums equal A003436, n >= 2, respectively A231622, n >= 1.

			The first few terms of the Taylor expansion of f(u; p) are:
f(u, p) = exp(p) * (1 + (1-2*p) * u^2/4 + (3-4*p+2*p^2) * u^4/16 + (15-18*p+9*p^2-2*p^3) * u^6/96 + (105-120*p+60*p^2-16*p^3+2*p^4) * u^8/768 + ... )
The first few rows of the T(n, k) triangle are:
n=0:     1
n=1:     1,     -2
n=2:     3,     -4,    2
n=3:    15,    -18,    9,    -2
n=4:   105,   -120,   60,   -16,   2
n=5:   945,  -1050,  525,  -150,  25,  -2
n=6: 10395, -11340, 5670, -1680, 315, -36, 2

J. W. Goodman, Introduction to Fourier Optics, 1996.
A. Papoulis, Systems and Transforms with Applications in Optics, 1968.

Andrew Howroyd, Rows n=0..50 of triangle, flattened
M. J. Bastiaans, The Wigner distribution function applied to optical signals and systems, Optics Communications, Vol. 25, nr. 1, pp. 26-30, 1978.
H. J. Butterweck, General theory of linear, coherent optical data processing systems, Journal of the Optical Society of America, Vol. 67, nr. 1, pp. 60-70, 1977.
J. W. Meijer, A note on optical diffraction, 1979.

Cf. A003436, A231622, A032184, A084534, A127674, A132062, A001497.
Cf. Related to the left hand columns: A001147, A001193, A261065.
Cf. Related to the right hand columns: A280560, A162395, A006011, A040977, A053347, A054334, A266561.

[[n le 0 select 1 else (-1)^k*2^(k-n+1)*Factorial(2*n-k-1)*Binomial(n, k)/Factorial(n-1): k in [0..n]]: n in [1..10]]; // G. C. Greubel, Nov 08 2018

T := proc(n, k): if n=0 then 1 else (-1)^k*2^(k-n+1)*n*(2*n-k-1)!/(k!*(n-k)!) fi: end: seq(seq(T(n, k), k=0..n), n=0..8);

Table[If[n==0 && k==0,1, (-1)^k*2^(k-n+1)*n*(2*n-k-1)!/(k!*(n-k)!)], {n, 0, 10}, {k,0,n}]//Flatten (* G. C. Greubel, Nov 08 2018 *)

T(n,k) = {if(n==0, 1, (-1)^k*2^(k-n+1)*n*(2*n-k-1)!/(k!*(n-k)!))}
for(n=0, 10, for(k=0, n, print1(T(n, k), ", ")); print); \\ Andrew Howroyd, Nov 08 2018

T(n, k) = (-1)^k*2^(k-n+1)*n*(2*n-k-1)!/(k!*(n-k)!), n > 0 and 0 <= k <= n, T(0, 0) = 1.
T(n, k) = (-1)^k*A001147(n-k)*A084534(n, k), n >= 0 and 0 <= k <= n.
T(n, k) = 2^(2*(k-n)+1)*A001147(n-k)*A127674(n, n-k), n > 0 and 0 <= k <= n, T(0, 0) = 1.
T(n, k) = (-1)^k*(A001497(n, k) + A132062(n, k)), n >= 1, T(0,0) = 1.

A342713 Partition the integers from 1 to n into three groups with consecutive numbers, then a(n) is the maximum value of the sum of the numbers in the second group multiplied by the minimum of the sum of the numbers in the first and third groups.

Original entry on oeis.org

2, 9, 21, 54, 90, 144, 234, 350, 504, 714, 950, 1350, 1764, 2156, 2772, 3500, 4374, 5390, 6380, 7812, 9504, 10890, 12740, 14850, 17442, 20475, 23100, 26334, 30444, 34320, 38709, 43146, 48510, 55250, 61047, 66780, 74925, 83600, 92169, 100485, 109350, 121512, 133331, 144000, 156195, 171171

Offset: 3

Scott R. Shannon, Mar 20 2021

nonn

The maximum product is obtained by making the sum of the numbers in the first and third groups as close as possible to each other and to half the sum of the numbers in the second group.

Geometrically the value of a(n) corresponds to the maximum area surrounded by three sides in a square-bottom 'U' shaped figure where the sides are drawn with single steps of incrementing length from 1 to n.

			a(3) = 2 as the only partition is {1},{2},{3}. The minimum sum of the first and third group is 1, thus a(3) = 2*1 = 2.
a(5) = 21 as the three group partition {1,2},{3,4},{5} has a minimum sum of the first and third groups of 1+2 = 3, thus a(5) = 3*(3+4) = 3*7 = 21.
a(12) = 714 as the three group partition {1,2,3,4,5,6},{7,8,9,10},{11,12} has a minimum sum of the first and third groups of 1+2+3+4+5+6 = 21, thus a(12) = 21*(7+8+9+10) = 21*34 = 714.

Cf. A000217, A006011, A007294, A008443, A286430.

cp's OEIS Frontend

A159572 A triangular array related to ordered partitions and having row sums 1,2,5,14,43,144,523,2048,8597... A047970.

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A305402 A number triangle T(n,k) read by rows for 0<=k<=n, related to the Taylor expansion of f(u, p) = (1/2)(1+1/(sqrt(1-u^2)))exp(p*sqrt(1-u^2)).

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A342713 Partition the integers from 1 to n into three groups with consecutive numbers, then a(n) is the maximum value of the sum of the numbers in the second group multiplied by the minimum of the sum of the numbers in the first and third groups.

Views

Author

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Comments

Examples

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cp's OEIS Frontend

A159572 A triangular array related to ordered partitions and having row sums 1,2,5,14,43,144,523,2048,8597... A047970.

Views

Author

Keywords

Comments

Crossrefs

Extensions

A305402 A number triangle T(n,k) read by rows for 0<=k<=n, related to the Taylor expansion of f(u, p) = (1/2)*(1+1/(sqrt(1-u^2)))*exp(p*sqrt(1-u^2)).

Views

Author

Keywords

Comments

Examples

References

Links

Crossrefs

Programs

Magma

Maple

Mathematica

PARI

Formula

A342713 Partition the integers from 1 to n into three groups with consecutive numbers, then a(n) is the maximum value of the sum of the numbers in the second group multiplied by the minimum of the sum of the numbers in the first and third groups.

Views

Author

Keywords

Comments

Examples

Crossrefs

A305402 A number triangle T(n,k) read by rows for 0<=k<=n, related to the Taylor expansion of f(u, p) = (1/2)(1+1/(sqrt(1-u^2)))exp(p*sqrt(1-u^2)).