A213778 -id:A213778 - cp's OEIS frontend

A213779 Principal diagonal of the convolution array A213778.

1, 6, 15, 33, 58, 97, 146, 214, 295, 400, 521, 671, 840, 1043, 1268, 1532, 1821, 2154, 2515, 2925, 3366, 3861, 4390, 4978, 5603, 6292, 7021, 7819, 8660, 9575, 10536, 11576, 12665, 13838, 15063, 16377, 17746, 19209, 20730, 22350, 24031, 25816, 27665, 29623

Offset: 1

Clark Kimberling, Jun 21 2012

Clark Kimberling, Table of n, a(n) for n = 1..1000
Index entries for linear recurrences with constant coefficients, signature (2,1,-4,1,2,-1).

Cf. A213778, A213500.

(See A213778.)
LinearRecurrence[{2,1,-4,1,2,-1},{1,6,15,33,58,97},80] (* Harvey P. Dale, Dec 12 2016 *)

Vec(x*(1+4*x+2*x^2+x^3)/((1-x)^4*(1+x)^2) + O(x^100)) \\ Colin Barker, Jan 31 2016

a(n) = 2*a(n-1) + a(n-2) - 4*a(n-3) + a(n-4) + 2*a(n-5) - a(n-6).
G.f.: x*(1+4*x+2*x^2+x^3) / ((1-x)^4*(1+x)^2).
From Colin Barker, Jan 31 2016: (Start)
a(n) = (16*n^3+30*n^2+2*(3*(-1)^n+7)*n+3*((-1)^n-1))/48.
a(n) = (8*n^3+15*n^2+10*n)/24 for n even.
a(n) = (8*n^3+15*n^2+4*n-3)/24 for n odd.
(End)

A213780 Antidiagonal sums of the convolution array A213778.

Original entry on oeis.org

1, 6, 17, 40, 78, 140, 230, 360, 535, 770, 1071, 1456, 1932, 2520, 3228, 4080, 5085, 6270, 7645, 9240, 11066, 13156, 15522, 18200, 21203, 24570, 28315, 32480, 37080, 42160, 47736, 53856, 60537, 67830, 75753, 84360, 93670, 103740, 114590

Offset: 1

Clark Kimberling, Jun 21 2012

Clark Kimberling, Table of n, a(n) for n = 1..1000
Index entries for linear recurrences with constant coefficients, signature (3,-1,-5,5,1,-3,1).

Cf. A213778, A213500.

Mathematica
```
(See A213778.)
```

a(n) = 3*a(n-1) - a(n-2) - 5*a(n-3) + 5*a(n-4) + a(n-5) - 3*a(n-6) + a(n-7).

G.f.: f(x)/g(x), where f(x) = x*(1 + 3*x) and g(x) = (1 + x)^2 *(1 - x)^5.

A213500 Rectangular array T(n,k): (row n) = bc, where b(h) = h, c(h) = h + n - 1, n >= 1, h >= 1, and = convolution.

Original entry on oeis.org

1, 4, 2, 10, 7, 3, 20, 16, 10, 4, 35, 30, 22, 13, 5, 56, 50, 40, 28, 16, 6, 84, 77, 65, 50, 34, 19, 7, 120, 112, 98, 80, 60, 40, 22, 8, 165, 156, 140, 119, 95, 70, 46, 25, 9, 220, 210, 192, 168, 140, 110, 80, 52, 28, 10, 286, 275, 255, 228, 196, 161, 125, 90

Offset: 1

Clark Kimberling, Jun 14 2012

Principal diagonal: A002412.
Antidiagonal sums: A002415.
Row 1:  (1,2,3,...)**(1,2,3,...) = A000292.
Row 2:  (1,2,3,...)**(2,3,4,...) = A005581.
Row 3:  (1,2,3,...)**(3,4,5,...) = A006503.
Row 4:  (1,2,3,...)**(4,5,6,...) = A060488.
Row 5:  (1,2,3,...)**(5,6,7,...) = A096941.
Row 6:  (1,2,3,...)**(6,7,8,...) = A096957.
...
In general, the convolution of two infinite sequences is defined from the convolution of two n-tuples:  let X(n) = (x(1),...,x(n)) and Y(n)=(y(1),...,y(n)); then X(n)**Y(n) = x(1)*y(n)+x(2)*y(n-1)+...+x(n)*y(1); this sum is the n-th term in the convolution of infinite sequences:(x(1),...,x(n),...)**(y(1),...,y(n),...), for all n>=1.
...
In the following guide to related arrays and sequences, row n of each array T(n,k) is the convolution b**c of the sequences b(h) and c(h+n-1). The principal diagonal is given by T(n,n) and the n-th antidiagonal sum by S(n). In some cases, T(n,n) or S(n) differs in offset from the listed sequence.
b(h)........ c(h)........ T(n,k) .. T(n,n) .. S(n)
h .......... h .......... A213500 . A002412 . A002415
h .......... h^2 ........ A212891 . A213436 . A024166
h^2 ........ h .......... A213503 . A117066 . A033455
h^2 ........ h^2 ........ A213505 . A213546 . A213547
h .......... h*(h+1)/2 .. A213548 . A213549 . A051836
h*(h+1)/2 .. h .......... A213550 . A002418 . A005585
h*(h+1)/2 .. h*(h+1)/2 .. A213551 . A213552 . A051923
h .......... h^3 ........ A213553 . A213554 . A101089
h^3 ........ h .......... A213555 . A213556 . A213547
h^3 ........ h^3 ........ A213558 . A213559 . A213560
h^2 ........ h*(h+1)/2 .. A213561 . A213562 . A213563
h*(h+1)/2 .. h^2 ........ A213564 . A213565 . A101094
2^(h-1) .... h .......... A213568 . A213569 . A047520
2^(h-1) .... h^2 ........ A213573 . A213574 . A213575
h .......... Fibo(h) .... A213576 . A213577 . A213578
Fibo(h) .... h .......... A213579 . A213580 . A053808
Fibo(h) .... Fibo(h) .... A067418 . A027991 . A067988
Fibo(h+1) .. h .......... A213584 . A213585 . A213586
Fibo(n+1) .. Fibo(h+1) .. A213587 . A213588 . A213589
h^2 ........ Fibo(h) .... A213590 . A213504 . A213557
Fibo(h) .... h^2 ........ A213566 . A213567 . A213570
h .......... -1+2^h ..... A213571 . A213572 . A213581
-1+2^h ..... h .......... A213582 . A213583 . A156928
-1+2^h ..... -1+2^h ..... A213747 . A213748 . A213749
h .......... 2*h-1 ...... A213750 . A007585 . A002417
2*h-1 ...... h .......... A213751 . A051662 . A006325
2*h-1 ...... 2*h-1 ...... A213752 . A100157 . A071238
2*h-1 ...... -1+2^h ..... A213753 . A213754 . A213755
-1+2^h ..... 2*h-1 ...... A213756 . A213757 . A213758
2^(n-1) .... 2*h-1 ...... A213762 . A213763 . A213764
2*h-1 ...... Fibo(h) .... A213765 . A213766 . A213767
Fibo(h) .... 2*h-1 ...... A213768 . A213769 . A213770
Fibo(h+1) .. 2*h-1 ...... A213774 . A213775 . A213776
Fibo(h) .... Fibo(h+1) .. A213777 . A001870 . A152881
h .......... 1+[h/2] .... A213778 . A213779 . A213780
1+[h/2] .... h .......... A213781 . A213782 . A005712
1+[h/2] .... [(h+1)/2] .. A213783 . A213759 . A213760
h .......... 3*h-2 ...... A213761 . A172073 . A002419
3*h-2 ...... h .......... A213771 . A213772 . A132117
3*h-2 ...... 3*h-2 ...... A213773 . A214092 . A213818
h .......... 3*h-1 ...... A213819 . A213820 . A153978
3*h-1 ...... h .......... A213821 . A033431 . A176060
3*h-1 ...... 3*h-1 ...... A213822 . A213823 . A213824
3*h-1 ...... 3*h-2 ...... A213825 . A213826 . A213827
3*h-2 ...... 3*h-1 ...... A213828 . A213829 . A213830
2*h-1 ...... 3*h-2 ...... A213831 . A213832 . A212560
3*h-2 ...... 2*h-1 ...... A213833 . A130748 . A213834
h .......... 4*h-3 ...... A213835 . A172078 . A051797
4*h-3 ...... h .......... A213836 . A213837 . A071238
4*h-3 ...... 2*h-1 ...... A213838 . A213839 . A213840
2*h-1 ...... 4*h-3 ...... A213841 . A213842 . A213843
2*h-1 ...... 4*h-1 ...... A213844 . A213845 . A213846
4*h-1 ...... 2*h-1 ...... A213847 . A213848 . A180324
[(h+1)/2] .. [(h+1)/2] .. A213849 . A049778 . A213850
h .......... C(2*h-2,h-1) A213853
...
Suppose that u = (u(n)) and v = (v(n)) are sequences having generating functions U(x) and V(x), respectively. Then the convolution u**v has generating function U(x)*V(x). Accordingly, if u and v are homogeneous linear recurrence sequences, then every row of the convolution array T satisfies the same homogeneous linear recurrence equation, which can be easily obtained from the denominator of U(x)*V(x). Also, every column of T has the same homogeneous linear recurrence as v.

			Northwest corner (the array is read by southwest falling antidiagonals):
  1,  4, 10, 20,  35,  56,  84, ...
  2,  7, 16, 30,  50,  77, 112, ...
  3, 10, 22, 40,  65,  98, 140, ...
  4, 13, 28, 50,  80, 119, 168, ...
  5, 16, 34, 60,  95, 140, 196, ...
  6, 19, 40, 70, 110, 161, 224, ...
T(6,1) = (1)**(6) = 6;
T(6,2) = (1,2)**(6,7) = 1*7+2*6 = 19;
T(6,3) = (1,2,3)**(6,7,8) = 1*8+2*7+3*6 = 40.

Clark Kimberling, Antidiagonals n = 1..60, flattened

Cf. A000027.

b[n_] := n; c[n_] := n
t[n_, k_] := Sum[b[k - i] c[n + i], {i, 0, k - 1}]
TableForm[Table[t[n, k], {n, 1, 10}, {k, 1, 10}]]
Flatten[Table[t[n - k + 1, k], {n, 12}, {k, n, 1, -1}]]
r[n_] := Table[t[n, k], {k, 1, 60}]  (* A213500 *)

t(n,k) = sum(i=0, k - 1, (k - i) * (n + i));
tabl(nn) = {for(n=1, nn, for(k=1, n, print1(t(k,n - k + 1),", ");); print(););};
tabl(12) \\ Indranil Ghosh, Mar 26 2017

def t(n, k): return sum((k - i) * (n + i) for i in range(k))
for n in range(1, 13):
    print([t(k, n - k + 1) for k in range(1, n + 1)]) # Indranil Ghosh, Mar 26 2017

T(n,k) = 4*T(n,k-1) - 6*T(n,k-2) + 4*T(n,k-3) - T(n,k-4).
T(n,k) = 2*T(n-1,k) - T(n-2,k).
G.f. for row n:  x*(n - (n - 1)*x)/(1 - x)^4.

A005744 Expansion of x(1+x-x^2)/((1-x)^4(1+x)).

Original entry on oeis.org

0, 1, 4, 9, 17, 28, 43, 62, 86, 115, 150, 191, 239, 294, 357, 428, 508, 597, 696, 805, 925, 1056, 1199, 1354, 1522, 1703, 1898, 2107, 2331, 2570, 2825, 3096, 3384, 3689, 4012, 4353, 4713, 5092, 5491, 5910, 6350, 6811, 7294, 7799, 8327, 8878, 9453, 10052

Offset: 0

N. J. A. Sloane, Simon Plouffe

Number of n-covers of a 2-set.
Boolean switching functions a(n,s) for s = 2.
Without the initial 0, this is row 1 of the convolution array A213778. - Clark Kimberling, Jun 21 2012
a(n) equals the second column of the triangle A355754. - Eric W. Weisstein, Mar 12 2024

R. J. Clarke, Covering a set by subsets, Discrete Math., 81 (1990), 147-152.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

T. D. Noe, Table of n, a(n) for n = 0..1000
Milan Janjić, Hessenberg Matrices and Integer Sequences, J. Int. Seq. 13 (2010) # 10.7.8.
A. V. Jayanthan, S. A. Seyed Fakhari, I. Swanson, and S. Yassemi, Induced matching, ordered matching and Castelnuovo-Mumford regularity of bipartite graphs, arXiv:2405.06781 [math.AC], 2024. See p. 17.
Vladeta Jovovic, Binary matrices up to row and column permutations.
Index entries for sequences related to Boolean functions
Index entries for linear recurrences with constant coefficients, signature (3,-2,-2,3,-1).

John W. Layman observes that A003453 appears to be the alternating sum transform (PSumSIGN) of A005744.
Cf. A002623, A005745, A005746, A005747, A005748, A005771, A003180, A008778, A052265.
Cf. A181971, A213778.
Cf. A355754.

CoefficientList[Series[x (1+x-x^2)/((1-x)^4(1+x)),{x,0,50}],x] (* or *) LinearRecurrence[{3,-2,-2,3,-1},{0,1,4,9,17},50] (* Harvey P. Dale, Apr 10 2012 *)

a(n)=([0,1,0,0,0; 0,0,1,0,0; 0,0,0,1,0; 0,0,0,0,1; -1,3,-2,-2,3]^n*[0;1;4;9;17])[1,1] \\ Charles R Greathouse IV, Feb 06 2017

a(n) = A002623(n) - (n+1).
a(n) = n*(n-1)/2 + Sum_{j=1..floor((n+1)/2)} (n-2*j+1)*(n-2*j)/2. - N. J. A. Sloane, Nov 28 2003
From R. J. Mathar, Apr 01 2010: (Start)
a(n) = 5*n/12 - 1/16 + 5*n^2/8 + n^3/12 + (-1)^n/16.
a(n) = 3*a(n-1) - 2*a(n-2) - 2*a(n-3) + 3*a(n-4) - a(n-5). (End)
a(n) = A181971(n+1, n-1) for n > 0. - Reinhard Zumkeller, Jul 09 2012
a(n) + a(n+1) = A008778(n). - R. J. Mathar, Mar 13 2021
E.g.f.: (x*(2*x^2 + 21*x + 27)*cosh(x) + (2*x^3 + 21*x^2 + 27*x - 3)*sinh(x))/24. - Stefano Spezia, Jul 27 2022

Additional comments from Alford Arnold

A213781 Rectangular array: (row n) = bc, where b(h) = 1+[h/2], c(h) = n-1+h, n>=1, h>=1, [ ] = floor, and = convolution.

Original entry on oeis.org

1, 4, 2, 9, 7, 3, 17, 14, 10, 4, 28, 25, 19, 13, 5, 43, 39, 33, 24, 16, 6, 62, 58, 50, 41, 29, 19, 7, 86, 81, 73, 61, 49, 34, 22, 8, 115, 110, 100, 88, 72, 57, 39, 25, 9, 150, 144, 134, 119, 103, 83, 65, 44, 28, 10, 191, 185, 173, 158, 138, 118, 94, 73, 49, 31

Offset: 1

Clark Kimberling, Jun 22 2012

Principal diagonal: A213782.
Antidiagonal sums: A005712.
row 1,  (1,2,2,3,3,4,4,...)**(1,2,3,4,5,6,7,...): A005744.
row 2,  (1,2,2,3,3,4,4,...)**(2,3,4,5,6,7,8,...).
row 3,  (1,2,2,3,3,4,4,...)**(3,4,5,6,7,8,9,...).
For a guide to related arrays, see A213500.

			Northwest corner (the array is read by falling antidiagonals):
1...4....9....17...28...43....62
2...7....14...25...39...58....81
3...10...19...33...50...73....100
4...13...24...41...61...88....119
5...16...29...49...72...103...138
6...19...34...57...83...118...157
7...22...39...65...94...133...176

Clark Kimberling, Antidiagonals n = 1..60, flattened

Cf. A213500, A213778.

b[n_] := Floor[(n + 2)/2]; c[n_] := n;
t[n_, k_] := Sum[b[k - i] c[n + i], {i, 0, k - 1}]
TableForm[Table[t[n, k], {n, 1, 10}, {k, 1, 10}]]
Flatten[Table[t[n - k + 1, k], {n, 12}, {k, n, 1, -1}]]
r[n_] := Table[t[n, k], {k, 1, 60}]  (* A213781 *)
s[n_] := Sum[t[i, n + 1 - i], {i, 1, n}]
s1 = Table[s[n], {n, 1, 50}] (* A005712 *)

T(n,k) = 3*T(n,k-1) - 2*T(n,k-2) - 2*T(n,k-3) + 3*T(n,k-4) - T(n,k-5).

G.f. for row n: f(x)/g(x), where f(x) = x*(n + x - (2*n - 1)*x^2 + (n -1)*x^3) and g(x) = (1 + x)(1 - x)^4.

cp's OEIS Frontend

A213779 Principal diagonal of the convolution array A213778.

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A213780 Antidiagonal sums of the convolution array A213778.

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A213500 Rectangular array T(n,k): (row n) = b**c, where b(h) = h, c(h) = h + n - 1, n >= 1, h >= 1, and ** = convolution.

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A005744 Expansion of x*(1+x-x^2)/((1-x)^4*(1+x)).

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A213781 Rectangular array: (row n) = b**c, where b(h) = 1+[h/2], c(h) = n-1+h, n>=1, h>=1, [ ] = floor, and ** = convolution.

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A213500 Rectangular array T(n,k): (row n) = bc, where b(h) = h, c(h) = h + n - 1, n >= 1, h >= 1, and = convolution.

A005744 Expansion of x(1+x-x^2)/((1-x)^4(1+x)).

A213781 Rectangular array: (row n) = bc, where b(h) = 1+[h/2], c(h) = n-1+h, n>=1, h>=1, [ ] = floor, and = convolution.