A104986
Matrix logarithm of triangle A104980.
Original entry on oeis.org
0, 1, 0, 2, 2, 0, 7, 4, 3, 0, 33, 14, 7, 4, 0, 191, 66, 27, 11, 5, 0, 1297, 382, 137, 48, 16, 6, 0, 10063, 2594, 843, 270, 79, 22, 7, 0, 87669, 20126, 6041, 1820, 495, 122, 29, 8, 0, 847015, 175338, 49219, 14176, 3679, 848, 179, 37, 9, 0, 8989301, 1694030, 448681, 124828, 31361, 6930, 1371, 252, 46, 10, 0
Offset: 0
Triangle begins:
0;
1, 0;
2, 2, 0;
7, 4, 3, 0;
33, 14, 7, 4, 0;
191, 66, 27, 11, 5, 0;
1297, 382, 137, 48, 16, 6, 0;
10063, 2594, 843, 270, 79, 22, 7, 0;
87669, 20126, 6041, 1820, 495, 122, 29, 8, 0;
847015, 175338, 49219, 14176, 3679, 848, 179, 37, 9, 0;
8989301, 1694030, 448681, 124828, 31361, 6930, 1371, 252, 46, 10, 0; ...
-
nmax = 10;
M = Table[If[n == k, 0, If[n == k+1, -n+1, -Coefficient[(1-1/Sum[i! x^i, {i, 0, n}])/x + O[x]^n, x, n-k-1]]], {n, 1, nmax+1}, {k, 1, nmax+1}];
T[n_, k_] /; 0 <= k <= n := Sum[(-1)^p MatrixPower[M, p][[n+1, k+1]]/p, {p, 1, n+1}]; T[, ] = 0;
Table[T[n, k], {n, 0, nmax}, {k, 0, n}] // Flatten (* Jean-François Alcover, Aug 09 2018, from PARI *)
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T(n,k)=if(n
A383450
2nd diagonal (from right) in A104978.
Original entry on oeis.org
2, 21, 180, 1430, 10920, 81396, 596904, 4326300, 31081050, 221760825, 1573537680, 11114897976, 78215948720, 548652722520, 3838040704080, 26784871943928, 186537501038070, 1296717366119175, 8999440181955300, 62366467037593950, 431633967218324640, 2983755440056831440, 20603495011611002400, 142131208489591604400
Offset: 0
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From Peter Luschny, May 04 2025: (Start)
a := n -> (3*n + 4)!/(n!*2*(2 + 2*n + 1)!): seq(a(n), n = 0..23);
gf := 2*hypergeom([5/3, 7/3], [5/2], (27*x)/4):
ser := series(gf, x, 25): seq(coeff(ser, x, k), k = 0..23); (End)
A383451
3nd diagonal (from right) in A104978.
Original entry on oeis.org
5, 84, 990, 10010, 92820, 813960, 6864396, 56241900, 450675225, 3548173200, 27536909400, 211183061544, 1603426948760, 12070359895440, 90193956545880, 669621798598200, 4943243777508855, 36308086251336900, 265483485367681350, 1933360478165412450, 14028103934595550800, 101447684961932268960, 731424072912190585200, 5258854714114889362800
Offset: 0
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From Peter Luschny, May 04 2025: (Start)
a := n -> (3*n + 6)!/(6*n!*(3 + 2*n + 1)!): seq(a(n), n = 0..23);
gf := 5*hypergeom([7/3, 8/3], [5/2], (27*x)/4):
ser := series(gf, x, 25): seq(coeff(ser, x, k), k = 0..23); (End)
Original entry on oeis.org
0, 0, 0, 12, 165, 1430, 10010, 61880, 352716, 1899240, 9806280, 49031400, 239028075, 1141710570, 5362579950, 24837212400, 113678010600, 515030986800, 2312957340720, 10307744670600, 45626928615450, 200758485907980, 878623171119540, 3826892034209552, 16596215454480200, 71691488703052400, 308585103547921200, 1323929637802371600
Offset: 0
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a := n -> ifelse(n < 3, 0, (3 + 2*n)! / (6*(n - 3)!*(n + 4)!)): seq(a(n), n = 0..27);
y := sqrt(1 - 4*x): gf := (1/(2*(x*y)^4))*((210*x^4 - 420*x^3 + 252*x^2 - 60*x + 5)/y -(32*x^4 - 176*x^3 + 162*x^2 - 50*x + 5)): ser := series(gf, x, 34):
seq(coeff(ser, x, n), n = 0..27); # Peter Luschny, May 04 2025
Showing 1-4 of 4 results.
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