A112500 Triangle of column sequences with a certain o.g.f. pattern.
1, 1, 1, 1, 4, 1, 1, 11, 10, 1, 1, 26, 60, 20, 1, 1, 57, 282, 225, 35, 1, 1, 120, 1149, 1882, 665, 56, 1, 1, 247, 4272, 13070, 9107, 1666, 84, 1, 1, 502, 14932, 79872, 100751, 35028, 3696, 120, 1, 1, 1013, 49996, 444902, 957197, 584325, 113428, 7470, 165, 1, 1
Offset: 0
Examples
Rows: [1]; [1,1]; [1,4,1]; [1,11,10,1]; [1,26,60,20,1]; [1,57,282,225,35,1]; ... a(4,3)= 60 = 6 + 12 + 9 + 12 + 9 + 12 from the binomial(4,2)=6 terms of the sum corresponding to (n_1,n_2,n_3) = (2,0,0), (0,2,0), (0,0,2), (1,1,0), (1,0,1) and (0,1,1).
Links
- W. Lang, First ten rows.
Crossrefs
Cf. A112501 (row sums).
Formula
G.f. column k: G(k, x):= x^(k-1)/product((1-j*x)^(k-j+1), j=1..k), k>=1.
a(n+k-1, k)=sum of product(binomial(n_j + k - 1, k - 1)*j^(n_j), j=1..k) with sum(n_j, j=1..k)=n, n_j >=0. There are binomial(n+k-1, k-1) terms of this sum and 1<=k<=n+1. a(n, k)=0 if n+1
A112498 Third column of second-order Eulerian triangle A008517 divided by 2.
3, 29, 164, 726, 2805, 9975, 33630, 109424, 347519, 1085313, 3349848, 10253994, 31203945, 94561643, 285716018, 861472836, 2593592883, 7800176565, 23441423340, 70410252350, 211411111133, 634610819679, 1904620987014
Offset: 3
Comments
See A004301 for the doubled sequence.
Links
- Index entries for linear recurrences with constant coefficients, signature (10,-40,82,-91,52,-12).
Crossrefs
Cf. A000295 (one half of second column).
A112504 Fifth column of triangle A112500.
1, 35, 665, 9107, 100751, 957197, 8110087, 62854845, 453710670, 3091406010, 20086835910, 125465290530, 758173316850, 4455503465430, 25571494599330, 143839855533270, 795332428661055, 4333564250230845, 23317657891319095
Offset: 0
Comments
For a combinatorial formula see A112500, case k=5.
Links
- Index entries for linear recurrences with constant coefficients, signature (35, -560, 5432, -35714, 168542, -589632, 1556776, -3126949, 4777591, -5506936, 4703032, -2881136, 1195632, -300672, 34560).
Programs
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Mathematica
CoefficientList[Series[1/Product[(1-j*x)^(6-j),{j,1,5}],{x,0,20}],x] (* Georg Fischer, Jul 10 2025 *)
Formula
G.f.: 1/product((1-j*x)^(6-j), j=1..5) = 1/(((1-x)^5)*((1-2*x)^4)*((1-3*x)^3)*((1-4*x)^2)*(1-5*x)).
a(n) computable from partial fraction decomposition of g.f. Cf. A112503.
Comments