A290315: Triangle T(n, k) read by rows: row n gives the coefficients of the numerator polynomials of the o.g.f. of the (n+1)-th diagonal of the Sheffer triangle A154537 (S2[2,1] generalized Stirling2), for n >= 0.

OEIS matches ?

• A290315: triangle, 55 terms agree, 10 full rows row-aligned (tabl) OEIS nonn in polytool catalog · view at the OEIS · generated b-file 990 terms from n = 0 ?

A generated b-file is computed from the node's definition along the row alignment of its OEIS link and starts at the entry's offset. It is an identification on a finite prefix, not a proof that the node and the OEIS entry agree for every n.

OEIS submission text ?

Fields to paste into an OEIS submission: the example %e, the formula %F and programs %t (Mathematica) and %o (PARI/GP) that list the same rows. For an OEIS link the row numbers, the offset and the orientation are those of the entry, as matched; the matched terms were compared with the published ones, which identifies the node with the entry on a finite prefix, not for every n. The PARI/GP program is not machine-checked here.

Triangle begins:
   1: 1;
   2: 1,     2;
   3: 1,    16,       12;
   4: 1,    66,      284,        120;
   5: 1,   224,     2872,       5952,        1680;
   6: 1,   706,    21080,     116336,      146064,        30240;
   7: 1,  2160,   132228,    1531072,     4804656,      4130304,       665280;
   8: 1,  6530,   760500,   16271080,   101422640,    208791648,    132557760,     17297280;
   9: 1, 19648,  4155120,  151922560,  1661273440,   6556459008,   9657333504,   4766423040,    518918400;
  10: 1, 59010, 21993776, 1304454880, 23155279200, 155184721088, 427142449920, 477104352768, 189945688320, 17643225600;

The row polynomials satisfy P(n,x) = (1+(8*n-14)*x)*P(n-1,x) + (2*x-4*x^2)*d/dx P(n-1,x) for n >= 2, with P(1,x) = 1.

P[1, x_] := 1;
P[n_, x_] := P[n, x] = Expand[(1 + (8*n - 14)*x)*P[n - 1, x] + (2*x - 4*x^2)*D[P[n - 1, x], x]];
Table[CoefficientList[P[n, x], x], {n, 1, 10}] // Flatten

P(N) = {my(v = vector(N)); v[1] = 1; for(n = 2, N, my(i = n); v[i] = (1 + (8*n - 14)*x)*v[i - 1] + (2*x - 4*x^2)*deriv(v[i - 1], x)); v};
row(n, p) = {my(c = Vecrev(p)); c};
T(N) = {my(v = P(N)); concat(vector(#v, i, row(i, v[i])))};
T(10)

Triangle begins:
  0: 1;
  1: 1,     2;
  2: 1,    16,       12;
  3: 1,    66,      284,        120;
  4: 1,   224,     2872,       5952,        1680;
  5: 1,   706,    21080,     116336,      146064,        30240;
  6: 1,  2160,   132228,    1531072,     4804656,      4130304,       665280;
  7: 1,  6530,   760500,   16271080,   101422640,    208791648,    132557760,     17297280;
  8: 1, 19648,  4155120,  151922560,  1661273440,   6556459008,   9657333504,   4766423040,    518918400;
  9: 1, 59010, 21993776, 1304454880, 23155279200, 155184721088, 427142449920, 477104352768, 189945688320, 17643225600;

The row polynomials satisfy P(n,x) = (1+(8*n-6)*x)*P(n-1,x) + (2*x-4*x^2)*d/dx P(n-1,x) for n >= 1, with P(0,x) = 1.

P[0, x_] := 1;
P[n_, x_] := P[n, x] = Expand[(1 + (8*n - 6)*x)*P[n - 1, x] + (2*x - 4*x^2)*D[P[n - 1, x], x]];
Table[PadRight[CoefficientList[P[n, x], x], n + 1], {n, 0, 9}] // Flatten

P(N) = {my(v = vector(N + 1)); v[1] = 1; for(n = 1, N, my(i = n + 1); v[i] = (1 + (8*n - 6)*x)*v[i - 1] + (2*x - 4*x^2)*deriv(v[i - 1], x)); v};
row(n, p) = {my(c = Vecrev(p)); c = concat(c, vector(n + 1 - #c)); c};
T(N) = {my(v = P(N)); concat(vector(#v, i, row(i - 1, v[i])))};
T(9)

Proved properties

• Real-rooted. Every row P_n, n >= 1, is zero or real-rooted with all roots in (-inf, 0)
  Every polynomial of the sequence has only real roots, inside the stated interval.
  by the cited steps: e-f3090409c727b3ad
• Coefficients of one sign. Every nonzero row, n >= 1: the coefficients of P_n have one sign
  All coefficients of each polynomial have the same sign (after x -> -x when alternating).
  derived from the cited steps by a stated theorem: e-f3090409c727b3ad
• Hadamard kernel (Malo). Rows P_n, n >= 1, are Hadamard kernels (preserve real-rootedness)
  Multiplying the coefficients of any real-rooted polynomial termwise by those of a row (Hadamard product) gives a real-rooted polynomial again.
  derived from the cited steps by a stated theorem: e-f3090409c727b3ad
• Interlacing of consecutive rows. Rows P_(m-1), P_m interlace (weak, previous interlaces next) for every m >= 2
  The roots of consecutive polynomials alternate on the real line.
  by the cited steps: e-f3090409c727b3ad

Proof chains

1 incoming proof step. Each step is expanded back to the roots of the proof DAG, the theorem checks of a node's own rule. JSON

• theorem_check liu_wang e-f3090409c727b3ad root: theorem check

  Liu–Wang theorem [LW07, Thm 2.1, Cor 2.4] ?

Used in 2 proof steps

• e-8644a9f1d4c74332 obreschkoff_combination → obreschkoff_combination of Triangle T(n, k) read by rows: row n gives the coefficients of…
• e-8644a9f1d4c74332 obreschkoff_combination → obreschkoff_combination of Triangle T(n, k) read by rows: row n gives the coefficients of…

Related sequences ?

No candidate equalities or related sequences.

References

1. [LW07] Lily L. Liu and Yi Wang. A unified approach to polynomial sequences with only real zeros. Advances in Applied Mathematics, 38(4):542–560, May 2007. doi:10.1016/j.aam.2006.02.003