A075497 → reverse

OEIS matches ?

• A004211: row sums, 15 terms agree OEIS nonn (row-sum match: finite agreement only) · view at the OEIS
• A075497: reversed triangle, 45 terms agree, 9 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: 1,  2;
  2: 1,  6,    4;
  3: 1, 12,   28,     8;
  4: 1, 20,  100,   120,     16;
  5: 1, 30,  260,   720,    496,      32;
  6: 1, 42,  560,  2800,   4816,    2016,      64;
  7: 1, 56, 1064,  8400,  27216,   30912,    8128,     128;
  8: 1, 72, 1848, 21168, 111216,  248640,  193600,   32640,    256;
  9: 1, 90, 3000, 47040, 365232, 1360800, 2182720, 1194240, 130816, 512;

The row polynomials satisfy P(n,x) = (1+2*n*x)*P(n-1,x) - 2*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 + 2*n*x)*P[n - 1, x] - 2*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] = (1 + 2*n*x)*v[i - 1] - 2*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:   2,      1;
   3:   4,      6,       1;
   4:   8,     28,      12,       1;
   5:  16,    120,     100,      20,       1;
   6:  32,    496,     720,     260,      30,      1;
   7:  64,   2016,    4816,    2800,     560,     42,     1;
   8: 128,   8128,   30912,   27216,    8400,   1064,    56,    1;
   9: 256,  32640,  193600,  248640,  111216,  21168,  1848,   72,  1;
  10: 512, 130816, 1194240, 2182720, 1360800, 365232, 47040, 3000, 90, 1;

Row n lists the coefficients of P(n,x) in descending powers of x, where P(n,x) = (1+(2*n-2)*x)*P(n-1,x) - 2*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 + (2*n - 2)*x)*P[n - 1, x] - 2*x^2*D[P[n - 1, x], x]];
Table[Reverse[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] = (1 + (2*n - 2)*x)*v[i - 1] - 2*x^2*deriv(v[i - 1], x)); v};
row(n, p) = {my(c = Vecrev(p)); c = concat(c, vector(n - #c)); c = Vecrev(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, 0]
  Every polynomial of the sequence has only real roots, inside the stated interval.
  by the cited steps: e-188635230e4781f5
• 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-188635230e4781f5
• 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-188635230e4781f5

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 reversal, shift 0 e-188635230e4781f5

  Affine, reversal and Möbius substitutions (elementary) polytool ?

  • premise (sequence): Stirling2 triangle with scaled diagonals (powers of 2).
    uses: Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, 0]
    • theorem_check liu_wang e-ca5f07049cdde3bf root: theorem check

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

Related sequences ?

Same fingerprint (1)

Equal up to content, sign, reversal and a small index shift.

• A075497: Stirling2 triangle with scaled diagonals (powers of 2).

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