A059374: Triangle read by rows, T(n, k) = Sum_{i=0..n} L'(n, n-i) * binomial(i, k), for k = 0..n-1.

OEIS matches ?

• A025168: row sums, 15 terms agree OEIS nonn (row-sum match: finite agreement only) · view at the OEIS
• A059374: triangle, 36 terms agree, 8 full rows row-aligned (tabl) OEIS nonn in polytool catalog · view at the OEIS · generated b-file 990 terms from n = 1 ?

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:
  0:        1;
  1:        3,         2;
  2:       13,        18,          6;
  3:       73,       156,        108,         24;
  4:      501,      1460,       1560,        720,        120;
  5:     4051,     15030,      21900,      15600,       5400,        720;
  6:    37633,    170142,     315630,     306600,     163800,      45360,      5040;
  7:   394353,   2107448,    4763976,    5891760,    4292400,    1834560,    423360,     40320;
  8:  4596553,  28393416,   75868128,  114335424,  106051680,   61810560,  22014720,   4354560,   362880;
  9: 58941091, 413689770, 1277703720, 2276043840, 2572547040, 1908930240, 927158400, 283046400, 48988800, 3628800;

The row polynomials satisfy P(n,x) = (2*n+1+2*n*x)*P(n-1,x) - (1+2*x+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[(2*n + 1 + 2*n*x)*P[n - 1, x] - (1 + 2*x + x^2)*D[P[n - 1, x], x]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten

P(N) = {my(v = vector(N + 1)); v[1] = 1; for(n = 1, N, my(i = n + 1); v[i] = (2*n + 1 + 2*n*x)*v[i - 1] - (1 + 2*x + 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 - 1, v[i])))};
T(9)

Triangle begins:
   1:        1;
   2:        3,         2;
   3:       13,        18,          6;
   4:       73,       156,        108,         24;
   5:      501,      1460,       1560,        720,        120;
   6:     4051,     15030,      21900,      15600,       5400,        720;
   7:    37633,    170142,     315630,     306600,     163800,      45360,      5040;
   8:   394353,   2107448,    4763976,    5891760,    4292400,    1834560,    423360,     40320;
   9:  4596553,  28393416,   75868128,  114335424,  106051680,   61810560,  22014720,   4354560,   362880;
  10: 58941091, 413689770, 1277703720, 2276043840, 2572547040, 1908930240, 927158400, 283046400, 48988800, 3628800;

The row polynomials satisfy P(n,x) = (2*n-1+(2*n-2)*x)*P(n-1,x) - (1+2*x+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[(2*n - 1 + (2*n - 2)*x)*P[n - 1, x] - (1 + 2*x + x^2)*D[P[n - 1, x], x]];
Table[PadRight[CoefficientList[P[n, x], x], n], {n, 1, 10}] // Flatten

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

Proved properties

• Real-rooted. Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, -1]
  Every polynomial of the sequence has only real roots, inside the stated interval.
  by the cited steps: e-cee0a9f908e74de4
• Coefficients of one sign. Every nonzero row, n >= 0: 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-cee0a9f908e74de4
• Hadamard kernel (Malo). Rows P_n, n >= 0, 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-cee0a9f908e74de4
• Interlacing of consecutive rows. Rows P_(m-1), P_m interlace (weak, unordered) for every m >= 1
  The roots of consecutive polynomials alternate on the real line.
  by the cited steps: e-cee0a9f908e74de4

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

• preserver_image affine, shift 0 e-cee0a9f908e74de4

  Affine, reversal and Möbius substitutions (elementary) polytool ?

  • premise (sequence): Triangular array A066667 or A008297 unsigned and transposed.
    uses: Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, 0]
    • theorem_check liu_wang e-db00758dc4aca02d root: theorem check

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

Used in 2 proof steps

• e-b96b8d87f5e7b835 obreschkoff_combination → obreschkoff_combination of Triangle read by rows, T(n, k) = Sum_{i=0..n} L'(n, n-i) * bin…
• e-b96b8d87f5e7b835 obreschkoff_combination → obreschkoff_combination of Triangle read by rows, T(n, k) = Sum_{i=0..n} L'(n, n-i) * bin…

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