A213548 -id:A213548 - cp's OEIS frontend

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

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.

A051836 a(n) = n(n+1)(n+2)(n+3)(3*n+2)/120.

Original entry on oeis.org

0, 1, 8, 33, 98, 238, 504, 966, 1716, 2871, 4576, 7007, 10374, 14924, 20944, 28764, 38760, 51357, 67032, 86317, 109802, 138138, 172040, 212290, 259740, 315315, 380016, 454923, 541198, 640088, 752928, 881144, 1026256, 1189881, 1373736, 1579641, 1809522, 2065414

Offset: 0

Barry E. Williams, Dec 12 1999

5-dimensional version of pentagonal-based pyramidal numbers. - Ben Creech (mathroxmysox(AT)yahoo.com)
If Y is a 3-subset of an n-set X then, for n>=7, a(n-6) is the number of 7-subsets of X having at least two elements in common with Y. - Milan Janjic, Nov 23 2007
Antidiagonal sums of the convolution array A213548. - Clark Kimberling, Jun 17 2012
After 0, convolution of nonzero triangular numbers (A000217) and nonzero pentagonal numbers (A000326). - Bruno Berselli, Jun 27 2013
a(n) is also the number of odd chordless cycles in the graph complement of the (n+1)-Andrásfai graph. - Eric W. Weisstein, Apr 14 2017

			By the fourth comment: A000217(1..6) and A000326(1..6) give the term a(6) = 1*21+5*15+12*10+22*6+35*3+51*1 = 504. - _Bruno Berselli_, Jun 27 2013

Albert H. Beiler, Recreations in the Theory of Numbers, Dover, N.Y., 1964, pp. 194-196.
Herbert John Ryser, Combinatorial Mathematics, "The Carus Mathematical Monographs", No. 14, John Wiley and Sons, 1963, pp. 1-8.

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Eric Weisstein's World of Mathematics, Andrásfai Graph.
Eric Weisstein's World of Mathematics, Chordless Cycle.
Eric Weisstein's World of Mathematics, Graph Complement.
Index to sequences related to pyramidal numbers.
Index entries for linear recurrences with constant coefficients, signature (6,-15,20,-15,6,-1).

Partial sums of A001296.
Cf. A093560 ((3, 1) Pascal, column m=5).
Cf. A000217, A000326, A213548.

[0] cat [Binomial(n+4, n)*(3*n+5)/5: n in [0..40]]; // Vincenzo Librandi, Jul 04 2017

with (combinat):a[0]:=0:for n from 1 to 50 do a[n]:=stirling2(n+2,n)+a[n-1] od: seq(a[n], n=0..34); # Zerinvary Lajos, Mar 17 2008

Table[n(n + 1)(n + 2)(n + 3)(3n + 2)/120, {n, 0, 60}] (* Vladimir Joseph Stephan Orlovsky, Apr 08 2011 *)
CoefficientList[Series[x (1 + 2 x) / (1 - x)^6, {x, 0, 33}], x] (* Vincenzo Librandi, Jul 04 2017 *)
LinearRecurrence[{6,-15,20,-15,6,-1},{0,1,8,33,98,238},40] (* Harvey P. Dale, Jun 01 2018 *)

a(n)=n*(n+1)*(n+2)*(n+3)*(3*n+2)/120 \\ Charles R Greathouse IV, Oct 07 2015

[((3*n+2)/(n+4))*binomial(n+4,5) for n in range(41)] # G. C. Greubel, Dec 27 2023

a(n) = C(n+4, n)*(3n+5)/5.
G.f.: x*(1+2*x)/(1-x)^6. (adapted by Vincenzo Librandi, Jul 04 2017)
From Amiram Eldar, Feb 15 2022: (Start)
Sum_{n>=1} 1/a(n) = 135*sqrt(3)*Pi/14 - 1215*log(3)/14 + 925/21.
Sum_{n>=1} (-1)^(n+1)/a(n) = 135*sqrt(3)*Pi/7 - 880*log(2)/7 - 355/21. (End)
E.g.f.: (1/5!)*x*(120 + 360*x + 240*x^2 + 50*x^3 + 3*x^4)*exp(x). - G. C. Greubel, Dec 27 2023

Simpler definition from Ben Creech (mathroxmysox(AT)yahoo.com), Nov 13 2005

A005718 Quadrinomial coefficients: C(2+n,n) + C(3+n,n) + C(4+n,n).

Original entry on oeis.org

3, 12, 31, 65, 120, 203, 322, 486, 705, 990, 1353, 1807, 2366, 3045, 3860, 4828, 5967, 7296, 8835, 10605, 12628, 14927, 17526, 20450, 23725, 27378, 31437, 35931, 40890, 46345, 52328, 58872, 66011, 73780, 82215, 91353, 101232, 111891, 123370, 135710, 148953, 163142, 178321, 194535

Offset: 0

N. J. A. Sloane

If Y is an (n-3)-subset of an n-set X then, for n>=5, a(n-5) is the number of 4-subsets of X having at least two elements in common with Y. - Milan Janjic, Dec 16 2007
This equation represents the number of numbers with <=n digits such that the sum of the digits is between 1 and 4 inclusive and no digit is larger than 3. - David Consiglio, Jr., Oct 27 2008
Row 2 of the convolution array A213548. - Clark Kimberling, Jun 20 2012

L. Comtet, Advanced Combinatorics, Reidel, 1974, p. 78.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Niall Byrnes, Gary R. W. Greaves, and Matthew R. Foreman, Bootstrapping cascaded random matrix models: correlations in permutations of matrix products, arXiv:2405.02541 [math-ph], 2024. See p. 7.
R. K. Guy, Letter to N. J. A. Sloane, 1987.
Simon Plouffe, Approximations de séries génératrices et quelques conjectures, Dissertation, Université du Québec à Montréal, 1992; arXiv:0911.4975 [math.NT], 2009.
Simon Plouffe, 1031 Generating Functions, Appendix to Thesis, Montreal, 1992.
Index entries for linear recurrences with constant coefficients, signature (5,-10,10,-5,1).

Cf. A008287, A062745, A063421, A071675, A213548.

[(((n+14)*n+71)*n+130)*n/24+3: n in [0..45]]; // Vincenzo Librandi, Jun 15 2011

A005718:=-(3-3*z+z**2)/(z-1)**5; # conjectured by Simon Plouffe in his 1992 dissertation

Table[Plus@@Table[Binomial[i + n, n], {i, 2, 4}], {n, 0, 43}] (* From Alonso del Arte, Jun 14 2011 *)

a(n)=(((n+14)*n+71)*n+130)*n/24+3 \\ Charles R Greathouse IV, Jun 14 2011

a(n) = binomial(n, 2)*(n^2+7*n+18)/12, n >= 2.
G.f.: (3-3*x+x^2)/(1-x)^5. (numerator polynomial is N4(4, x) from A063421).
a(n) = A008287(n, 4), n >= 2 (fifth column of quadrinomial coefficients).
a(n) = A062745(n, 4), n >= 2 (fifth column).
a(n) = 3*C(n+2,2) + 3*C(n+2,3) + C(n+2,4) (see comment in A071675). - Vladimir Shevelev and Peter J. C. Moses, Jun 22 2012
E.g.f.: exp(x)*(72 + 216*x + 120*x^2 + 20*x^3 + x^4)/24. - Stefano Spezia, May 09 2024

Better description from Zerinvary Lajos, Dec 02 2005

A213550 Rectangular array: (row n) = b**c, where b(h) = h*(h+1)/2, c(h) = n-1+h, n>=1, h>=1, and ** = convolution.

Original entry on oeis.org

1, 5, 2, 15, 9, 3, 35, 25, 13, 4, 70, 55, 35, 17, 5, 126, 105, 75, 45, 21, 6, 210, 182, 140, 95, 55, 25, 7, 330, 294, 238, 175, 115, 65, 29, 8, 495, 450, 378, 294, 210, 135, 75, 33, 9, 715, 660, 570, 462, 350, 245, 155, 85, 37, 10, 1001, 935, 825, 690, 546

Offset: 1

Clark Kimberling, Jun 16 2012

Principal diagonal:  A002418
Antidiagonal sums:  A005585
row 1,  (1,3,6,...)**(1,2,3,...):  A000332
row 2,  (1,3,6,...)**(2,3,4,...):  A005582
row 3,  (1,3,6,...)**(3,4,5,...):  A095661
row 4,  (1,3,6,...)**(4,5,6,...):  A095667
For a guide to related arrays, see A213500.

			Northwest corner (the array is read by falling antidiagonals):
1....5....15...35....70....126
2....9....25...55....105...182
3....13...35...75....140...238
4....17...45...95....175...294
5....21...55...115...210...350

Cf. A213500, A213548.

b[n_] := n (n + 1)/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}]  (* A213550 *)
d = Table[t[n, n], {n, 1, 40}] (* A002418 *)
s[n_] := Sum[t[i, n + 1 - i], {i, 1, n}]
s1 = Table[s[n], {n, 1, 50}] (* A005585 *)

T(n,k) = 5*T(n,k-1) - 10*T(n,k-2) + 10*T(n,k-3) - 5*T(n,k-4) + T(n,k-5).

G.f. for row n: f(x)/g(x), where f(x) = n-(n-1)*x and g(x) = (1 - x)^5.

cp's OEIS Frontend

A213549 Principal diagonal of the convolution array A213548.

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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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A051836 a(n) = n*(n+1)*(n+2)*(n+3)*(3*n+2)/120.

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A005718 Quadrinomial coefficients: C(2+n,n) + C(3+n,n) + C(4+n,n).

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A213550 Rectangular array: (row n) = b**c, where b(h) = h*(h+1)/2, c(h) = n-1+h, n>=1, h>=1, 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.

A051836 a(n) = n(n+1)(n+2)(n+3)(3*n+2)/120.