A033455 -id:A033455 - cp's OEIS frontend

A213503 Rectangular array: (row n) = bc, where b(h) = h^2, c(h) = n-1+h, n>=1, h>=1, and = convolution.

1, 6, 2, 20, 11, 3, 50, 34, 16, 4, 105, 80, 48, 21, 5, 196, 160, 110, 62, 26, 6, 336, 287, 215, 140, 76, 31, 7, 540, 476, 378, 270, 170, 90, 36, 8, 825, 744, 616, 469, 325, 200, 104, 41, 9, 1210, 1110, 948, 756, 560, 380, 230, 118, 46, 10

Offset: 1

Clark Kimberling, Jun 16 2012

Principal diagonal:  A117066
Antidiagonal sums:  A033455
For a guide to related arrays, see A213500.

			Northwest corner (the array is read by falling antidiagonals):
1....6....20....50....105....196...336
2....11...34....80....160....287...476
3....16...48....110...215....378...616
4....21...62....140...270....469...756
5....26...76....170...325....560...896
...
T(5,1) = (1)**(5) = 5
T(5,2) = (1,4)**(5,6) = 1*6+4*5 = 26
T(5,3) = (1,4,9)**(5,6,7) = 1*7+4*6+9*5 = 76

G. C. Greubel, Antidiagonal rows n = 1..100

Cf. A213500.

Flat(List([1..12], n-> List([1..n], k-> (n-k+1)*((n-k+1)^3 + 4*(n-k+1)^2*k + 6*k*(n-k+1) - n + 3*k - 1)/12))); # G. C. Greubel, Jul 05 2019

[[(n-k+1)*((n-k+1)^3 + 4*(n-k+1)^2*k + 6*k*(n-k+1) - n + 3*k - 1)/12: k in [1..n]]: n in [1..12]]; // G. C. Greubel, Jul 05 2019

(* First program *)
b[n_]:= n^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}]] (* A213503 *)
r[n_]:= Table[T[n, k], {k,40}]  (* columns of antidiagonal triangle *)
d = Table[T[n, n], {n, 1, 40}] (* A117066 *)
s[n_]:= Sum[T[i, n+1-i], {i, 1, n}]
s1 = Table[s[n], {n, 1, 50}] (* A033455 *)
(* Second program *)
Table[(n-k+1)*((n-k+1)^3 + 4*(n-k+1)^2*k + 6*k*(n-k+1) - n + 3*k - 1)/12, {n, 12}, {k, n}]//Flatten (* G. C. Greubel, Jul 05 2019 *)

t(n,k) = (n-k+1)*((n-k+1)^3 + 4*(n-k+1)^2*k + 6*k*(n-k+1) - n + 3*k - 1)/12;
for(n=1,12, for(k=1,n, print1(t(n,k), ", "))) \\ G. C. Greubel, Jul 05 2019

[[(n-k+1)*((n-k+1)^3 + 4*(n-k+1)^2*k + 6*k*(n-k+1) - n + 3*k - 1)/12 for k in (1..n)] for n in (1..12)] # G. C. Greubel, Jul 05 2019

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 + x - (n - 1)^2 x^2 and g(x) = (1 - x)^5.
T(n,k) = k*(k^3 + 4*k^2*n + 6*k*n - k + 2*n)/12. - G. C. Greubel, Jul 05 2019

A247645 Triangle read by rows: T(j,0)=1, T(0,j) = [j=0], T(-1,j)=T(-2,j)=0, T(j,k)=2T(j-1,k)-T(j-2,k)+T(j-1,k-2)+T(j-2,k-2).

Original entry on oeis.org

1, 1, 1, 1, 1, 2, 4, 1, 1, 1, 3, 9, 5, 7, 1, 1, 1, 4, 16, 14, 26, 8, 10, 1, 1, 1, 5, 25, 30, 70, 34, 52, 11, 13, 1, 1, 1, 6, 36, 55, 155, 104, 190, 63, 87, 14, 16, 1, 1, 1, 7, 49, 91, 301, 259, 553, 253, 403, 101, 131, 17, 19, 1, 1, 1, 8, 64, 140, 532, 560

Offset: 0

N. J. A. Sloane, Sep 23 2014

			Triangle begins:
1,
1,1,1,
1,2,4,1,1,
1,3,9,5,7,1,1,
1,4,16,14,26,8,10,1,1,
1,5,25,30,70,34,52,11,13,1,1,
1,6,36,55,155,104,190,63,87,14,16,1,1
1,7,49,91,301,259,553,253,403,101,131,17,19,1,1
1,8,64,140,532,560,1372,806,1462,504,736,148,184,20,22,1,1
...

Lars Blomberg, Table of n, a(n) for n = 0..1680 (rows 0-40)
Johann Cigler, Some remarks and conjectures related to lattice paths in strips along the x-axis, arXiv:1501.04750 [math.CO], 2015-2016.

Column 0-5 is A000012, A000027, A000290, A000330, A006325, A033455.

T[, 0] = 1; T[j, k_] /; 0 <= k <= 2j := T[j, k] = Which[k<0 || k>2j, 0, k == 2j || k == 2j-1, 1, OddQ[k], T[j-1, k] + T[j-1, k-1], EvenQ[k], T[j-1, k-2] + 2 T[j-1, k-1] + T[j-1, k]];
Table[T[j, k], {j, 0, 8}, {k, 0, 2j}] // Flatten (* Jean-François Alcover, Oct 09 2018 *)

More terms from Lars Blomberg, Aug 05 2015

A271663 Convolution of nonzero squares (A000290) with nonzero pentagonal numbers (A000326).

Original entry on oeis.org

1, 9, 41, 131, 336, 742, 1470, 2682, 4587, 7447, 11583, 17381, 25298, 35868, 49708, 67524, 90117, 118389, 153349, 196119, 247940, 310178, 384330, 472030, 575055, 695331, 834939, 996121, 1181286, 1393016, 1634072, 1907400, 2216137, 2563617, 2953377, 3389163, 3874936

Offset: 0

Ilya Gutkovskiy, Apr 12 2016

More generally, the ordinary generating function for the convolution of nonzero h-gonal numbers and k-gonal numbers is (1 + (h - 3)*x)*(1 + (k - 3)*x)/(1 - x)^6.

G. C. Greubel, Table of n, a(n) for n = 0..1000
OEIS Wiki, Figurate numbers
Eric Weisstein's World of Mathematics, Square Number
Eric Weisstein's World of Mathematics, Pentagonal Number
Index entries for linear recurrences with constant coefficients, signature (6,-15,20,-15,6,-1)

Cf. A000290, A000326.
Cf. A005585: convolution of nonzero squares with nonzero triangular numbers.
Cf. A033455: convolution of nonzero squares with themselves.
Cf. A051836 (after 0): convolution of nonzero triangular numbers with nonzero pentagonal numbers.

/* From definition: */ P:=func; /*, where P(n,k) is the n-th k-gonal number, */ [&+[P(n+1-i,4)*P(i,5): i in [1..n]]: n in [1..40]]; // Bruno Berselli, Apr 12 2016

[(n+1)*(n+2)*(n+3)*(6*n^2+19*n+20)/120: n in [0..40]]; // Bruno Berselli, Apr 12 2016

LinearRecurrence[{6, -15, 20, -15, 6, -1}, {1, 9, 41, 131, 336, 742}, 40]
Table[(n + 1) (n + 2) (n + 3) (6 n^2 + 19 n + 20)/120, {n, 0, 40}]
With[{nmax = 50}, CoefficientList[Series[(120 + 960*x + 1440*x^2 + 680*x^3 + 115*x^4 + 6*x^5)*Exp[x]/120, {x, 0, nmax}], x]*Range[0, nmax]!] (* G. C. Greubel, Jun 07 2017 *)

vector(40, n, n--; (n+1)*(n+2)*(n+3)*(6*n^2+19*n+20)/120) \\ Altug Alkan, Apr 12 2016

O.g.f.: (1 + x)*(1 + 2*x)/(1 - x)^6.
E.g.f.: (120 + 960*x + 1440*x^2 + 680*x^3 + 115*x^4 + 6*x^5)*exp(x)/120.
a(n) = 6*a(n-1) - 15*a(n-2) + 20*a(n-3) - 15*a(n-4) + 6*a(n-5) - a(n-6).
a(n) = (n + 1)*(n + 2)*(n + 3)*(6*n^2 + 19*n + 20)/120.
Sum_{n>=0} 1/a(n) = 1.149165731...

Edited by Bruno Berselli, Apr 12 2016

A306548 Triangle T(n,k) read by rows, where the k-th column is the shifted self-convolution of the power function n^k, n >= 0, 0 <= k <= n.

Original entry on oeis.org

0, 0, 0, 1, 0, 0, 2, 1, 0, 0, 3, 4, 1, 0, 0, 4, 10, 8, 1, 0, 0, 5, 20, 34, 16, 1, 0, 0, 6, 35, 104, 118, 32, 1, 0, 0, 7, 56, 259, 560, 418, 64, 1, 0, 0, 8, 84, 560, 2003, 3104, 1510, 128, 1, 0, 0, 9, 120, 1092, 5888, 16003, 17600, 5554, 256, 1, 0, 0, 10, 165, 1968, 14988, 64064, 130835, 101504, 20758, 512, 1, 0, 0

Offset: 0

Kolosov Petro, Feb 23 2019

For n > 0 an odd-power identity n^(2m+1)+1, m >= 0 can be found using the current sequence. The sum of the n-th diagonal of T(n,k) over 0 <= k <= m multiplied by A(m,k) gives n^(2m+1)-1, where A(m,k) = A302971(m,k)/A304042(m,k). For example, consider the case n=4, m=2: the n-th diagonal of T(n, 0 <= k <= m) is {5, 10, 34}, and the m-th row of triangle A(m, 0 <= k <= m) is {1, 0, 30}, thus (3+1)^5 + 1 = 5*1 + 10*0 + 34*30 = 1025.

			==================================================================
k=    0     1     2     3      4      5     6    7    8    9    10
==================================================================
n=0:  2;
n=1:  2,    0;
n=2:  3,    0,    0;
n=3:  4,    1,    0,    0;
n=4:  5,    4,    1,    0,     0;
n=5:  6,   10,    8,    1,     0,     0;
n=6:  7,   20,   34,   16,     1,     0,    0;
n=7:  8,   35,  104,  118,    32,     1,    0,   0;
n=8:  9,   56,  259,  560,   418,    64,    1,   0,   0;
n=9:  10,  84,  560, 2003,  3104,  1510,  128,   1,   0,   0;
n=10: 11, 120, 1092, 5888, 16003, 17600, 5554, 256,   1,   0;   0;
...

D. V. Widder et al., The Convolution Transform, Bull. Amer. Math. Soc. 60 (1954), 444-456.
Wikipedia, Convolution.
Wikipedia, Convolution power.

Nonzero terms of columns k=0..5 give: A000027, A000292, A033455, A145216, A145217, A145218.
Partial sums of columns k=1..2 give: A000332, A259181.
Cf. A302971, A304042, A198633, A000079.

f[m_, s_] := Piecewise[{{s^m, s >= 0}, {0, True}}];
F[n_, m_] := Sum[f[m, n - k]*f[m, k], {k, -Infinity, +Infinity}];
T[n_, k_] := F[n - k, k];
Column[Table[T[n, k], {n, 0, 12}, {k, 0, n}], Left]

f(m, s) = s^m, if s >= 0;
f(m, s) = 0, otherwise.
F(n,m) = Sum_{k} f(m, n-k) * f(m, k), -oo < k < +oo;
T(n,k) = F(n-k, k).

Edited by Kolosov Petro, Mar 13 2019

A339355 Maximum number of copies of a 12345 permutation pattern in an alternating (or zig-zag) permutation of length n + 7.

Original entry on oeis.org

8, 16, 64, 112, 272, 432, 832, 1232, 2072, 2912, 4480, 6048, 8736, 11424, 15744, 20064, 26664, 33264, 42944, 52624, 66352, 80080, 99008, 117936, 143416, 168896, 202496, 236096, 279616, 323136, 378624, 434112, 503880, 573648, 660288, 746928, 853328, 959728, 1089088, 1218448

Offset: 1

Lara Pudwell, Dec 01 2020

nonn

The maximum number of copies of 123 in an alternating permutation is motivated in the Notices reference, and the argument here is analogous.

			a(1) = 8. The alternating permutation of length 1 + 7 = 8 with the maximum number of copies of 12345 is 13254768. The eight copies are 12468, 12478, 12568, 12578, 13468, 13478, 13568, and 13578.

Georg Fischer, Table of n, a(n) for n = 1..200
Lara Pudwell, From permutation patterns to the periodic table, Notices of the American Mathematical Society. 67.7 (2020), 994-1001.

Cf. A005585, A033455, A168380.

a := proc(n2) local n; n:= floor(n2/2): if n2 = 2*n then 32*binomial(n+4,5) - 16*binomial(n+3,4) else n:=n+1; (4*n*(n^4+5*n^3+10*n^2+10*n+4))/15 fi end; seq(a(n), n=1..20); # Georg Fischer, Nov 25 2022

a(2*n) = 16*A005585(n) = 32*binomial(n+4, 5) - 16*binomial(n+3, 4).
a(2*n-1) = 8*A033455(n) = (4*n*(n^4 + 5*n^3 + 10*n^2 + 10*n + 4))/15.
D-finite with recurrence: (n-1)*((n-3)^2+9*n-6)*a(n) - (2*(n-3)^2+20*n-16)*a(n-1) - (n+5)*((n-3)^2+11*n-2)*a(n-2) = 0. - Georg Fischer, Nov 25 2022

A195166 Numbers expressible as 2^a - 2^b, with 0 <= b < a, such that n^a - n^b is divisible by 2^a - 2^b for all n.

Original entry on oeis.org

1, 2, 6, 12, 30, 24, 60, 120, 252, 240, 504, 16380, 32760, 65520

Offset: 1

Michel Marcus, Dec 21 2012

= 2^1  - 2^0. (n^1  - n^0)/1     : A000027
= 2^2  - 2^1. (n^2  - n^1)/2     : A000217
= 2^3  - 2^1. (n^3  - n^1)/6     : A000292
= 2^4  - 2^2. (n^4  - n^2)/12    : A002415
= 2^5  - 2^1. (n^5  - n^1)/30    : A033455
= 2^5  - 2^3. (n^5  - n^3)/24    : A006414
= 2^6  - 2^2. (n^6  - n^2)/60    : A213547
= 2^7  - 2^3. (n^7  - n^3)/120   : A114239
= 2^8  - 2^2. (n^8  - n^2)/252   :
= 2^8  - 2^4. (n^8  - n^4)/240   : A078876
= 2^9  - 2^3. (n^9  - n^3)/504   :
= 2^14 - 2^2. (n^14 - n^2)/16380 :
= 2^15 - 2^3. (n^15 - n^3)/32760 :
= 2^16 - 2^4. (n^16 - n^4)/65520 :
Comment from Tomohiro Yamada, Oct 05 2022: (Start)
"The Mod Set Stanford University" and Carl Pomerance independently noted that the completeness of this sequence follows from a result of Schinzel on primitive prime factors of sequences a^n - b^n in the remark to a problem of Harry Ruderman asking whether if 2^a - 2^b divides 3^a - 3^b, then 2^a - 2^b belongs to this sequence.
The Mod Set verified that if a > b, 2^a - 2^b divides 3^a - 3^b, but 2^a - 2^b does not belong to this sequence, then a - b > 1900. Ram Murty and Kumar Murty proved that there are only finitely many natural numbers a, b such that 2^a - 2^b divides 3^a - 3^b.
Qi Sun and Ming Zhi Zhang also showed that if a > b and n^a - n^b is divisible by 2^a - 2^b for all n, then 2^a - 2^b belongs to this sequence. (End)

			a(3) = 6 belongs to this sequence since (n^3 - n)/6 = C(n+1, 3) = A000292(n-1).

M. Ram Murty and V. Kumar Murty, On a Problem of Ruderman, Amer. Math. Monthly 118 (2011), 644-650, available from the first author's website.
Harry Ruderman, Problem E2468, Amer. Math. Monthly 81 (1974), p. 405.
A. Schinzel, On primitive prime factors of a^n - b^n, Proc. Cambridge Phil. Soc. 58 (1962), 556-562.
Qi Sun and Ming Zhi Zhang, Pairs where 2^a-2^b divides n^a-n^b for all n, Proc. Amer. Math. Soc. 93 (1985), 218-220.
The Mod Set Stanford University and Carl Pomerance, When 2^m - 2^n divides 3^m - 3^n, remarks to Problem E2468*, Amer. Math. Monthly 84 (1977), 59-60.
W. Y. Velez, When 2^m - 2^n divides 3^m - 3^n, remarks to Problem E2468, Amer. Math. Monthly 83 (1976), 288-289.

cp's OEIS Frontend

A213503 Rectangular array: (row n) = b**c, where b(h) = h^2, c(h) = n-1+h, n>=1, h>=1, and ** = convolution.

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A247645 Triangle read by rows: T(j,0)=1, T(0,j) = [j=0], T(-1,j)=T(-2,j)=0, T(j,k)=2T(j-1,k)-T(j-2,k)+T(j-1,k-2)+T(j-2,k-2).

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A271663 Convolution of nonzero squares (A000290) with nonzero pentagonal numbers (A000326).

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Magma

Magma

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A306548 Triangle T(n,k) read by rows, where the k-th column is the shifted self-convolution of the power function n^k, n >= 0, 0 <= k <= n.

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A339355 Maximum number of copies of a 12345 permutation pattern in an alternating (or zig-zag) permutation of length n + 7.

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Maple

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A195166 Numbers expressible as 2^a - 2^b, with 0 <= b < a, such that n^a - n^b is divisible by 2^a - 2^b for all n.

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A213503 Rectangular array: (row n) = bc, where b(h) = h^2, c(h) = n-1+h, n>=1, h>=1, and = convolution.