A077170 -id:A077170 - cp's OEIS frontend

A077169 Initial terms of rows of A077168.

1, 2, 4, 7, 11, 15, 20, 26, 33, 41, 50, 60, 71, 83, 96, 110, 125, 141, 158, 176, 195, 215, 236, 258, 281, 305, 330, 356, 383, 411, 440, 470, 501, 533, 566, 600, 635, 671, 708, 746, 785, 825, 866, 908, 951, 995, 1040, 1086, 1133, 1181, 1230, 1280, 1331, 1383

Offset: 0

Amarnath Murthy, Nov 01 2002

G. C. Greubel, Table of n, a(n) for n = 0..5000
Index entries for linear recurrences with constant coefficients, signature (3,-3,1).

Cf. A077168, A077170.

Join[{1, 2, 4, 7}, Table[(n^2 - n + 10)/2, {n, 4, 50}]] (* G. C. Greubel, Jul 13 2017 *)

concat([1,2,4,7], for(n=4,50, print1((n^2 - n +10)/2, ", "))) \\ G. C. Greubel, Jul 13 2017

For n>3, a(n) = (n^2-n+10)/2.
From G. C. Greubel, Jul 13 2017: (Start)
a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3), n >= 7.
G.f.: (1-x+x^2-x^5 + x^6)/(1-x)^3.
E.g.f.: (1/2)*(x^2 + 10)*exp(x) - 4 - 3*x + x^2 - x^3/6. (End)
Sum_{n>=0} 1/a(n) = 1177/840 + 2*Pi*tanh(sqrt(39)*Pi/2)/sqrt(39). - Amiram Eldar, Dec 13 2022

More terms from Sascha Kurz, Feb 10 2003

A077168 Lexicographically earliest infinite sequence of distinct positive numbers with the property that when written as a triangle, the product of each row is a factorial.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 259200, 15, 16, 17, 18, 19, 87178291200, 20, 21, 22, 23, 24, 25, 202741834014720, 26, 27, 28, 29, 30, 31, 32, 484725313854093312000000, 33, 34, 35, 36, 37, 38, 39, 40, 4438779300500903005519872000000, 41, 42, 43, 44

Offset: 0

Amarnath Murthy, Nov 01 2002

The old definition was "Triangle formed by grouping the natural numbers so that the n-th group contains n numbers whose product is a factorial.". - N. J. A. Sloane, Oct 06 2024

			Triangle begins:
1,
2, 3,
4, 5, 6,
7, 8, 9, 10,
11, 12, 13, 14, 259200,
15, 16, 17, 18, 19, 87178291200,
20, 21, 22, 23, 24, 25, 202741834014720,
26, 27, 28, 29, 30, 31, 32, 484725313854093312000000,
33, 34, 35, 36, 37, 38, 39, 40, 4438779300500903005519872000000,
...
The row products are:
 1 = 1!
 2*3 = 6 = 3!
 4*5*6 = 120 = 5!
 7*8*9*10 = 5040 = 7!
 11*12*13*14*259200 = 6227020800 = 13!
 15*16*17*18*19*87178291200 = 121645100408832000 = 19!
 20*21*22*23*24*25*202741834014720 = 25852016738884976640000 = 23!
 26*27*28*29*30*31*32*484725313854093312000000 = 8222838654177922817725562880000000 = 31!
 33*34*35*36*37*38*39*40*4438779300500903005519872000000 = 37!
 ...

Cf. A076031, A077169, A077170, A077171.

More terms from Sascha Kurz, Feb 10 2003

Entry revised by N. J. A. Sloane, Oct 06 2024

A077171 a(n) = k where k! is the product of the terms of the n-th row of A077168.

Original entry on oeis.org

1, 3, 5, 7, 13, 19, 23, 31, 37, 47, 59, 67, 79, 89, 109, 113, 139, 157, 173, 193, 211, 233, 257, 277, 293, 317, 353, 379, 409, 439, 467, 499, 523, 563, 599, 631, 661, 701, 743, 773, 823, 863, 907, 947, 991, 1039, 1069, 1129, 1171, 1229, 1279, 1327, 1381, 1433

Offset: 0

Amarnath Murthy, Nov 01 2002

nonn

Cf. A077168, A077169, A077170.

More terms from Sascha Kurz, Feb 10 2003

Deleted an ambiguous comment and an incorrect comment, at the suggestion of Harvey P. Dale. - N. J. A. Sloane, Oct 06 2024

cp's OEIS Frontend

A077169 Initial terms of rows of A077168.

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A077168 Lexicographically earliest infinite sequence of distinct positive numbers with the property that when written as a triangle, the product of each row is a factorial.

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A077171 a(n) = k where k! is the product of the terms of the n-th row of A077168.

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