A185410: A decomposition of the double factorials A001147.

OEIS matches ?

• A001147: row sums, 15 terms agree OEIS nonn (row-sum match: finite agreement only) · view at the OEIS · generated b-file 200 terms from n = 0 ?
• A156919: reversed triangle, 39 terms agree, 8 full rows row-aligned (tabl) OEIS nonn · 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:
  0: 1;
  1: 1;
  2: 1,    2;
  3: 1,   10,      4;
  4: 1,   36,     60,       8;
  5: 1,  116,    516,     296,       16;
  6: 1,  358,   3508,    5168,     1328,       32;
  7: 1, 1086,  21120,   64240,    42960,     5664,      64;
  8: 1, 3272, 118632,  660880,   900560,   320064,   23488,   128;
  9: 1, 9832, 638968, 6049744, 14713840, 10725184, 2225728, 95872, 256;

The row polynomials satisfy P(n,x) = (1+(2*n-2)*x)*P(n-1,x) + (2*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 - 2)*x)*P[n - 1, x] + (2*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 - 2)*x)*v[i - 1] + (2*x - 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:
  0:   1;
  1:   2,      1;
  2:   4,     10,        1;
  3:   8,     60,       36,         1;
  4:  16,    296,      516,       116,         1;
  5:  32,   1328,     5168,      3508,       358,         1;
  6:  64,   5664,    42960,     64240,     21120,      1086,        1;
  7: 128,  23488,   320064,    900560,    660880,    118632,     3272,       1;
  8: 256,  95872,  2225728,  10725184,  14713840,   6049744,   638968,    9832,     1;
  9: 512, 387840, 14792448, 114735936, 265115424, 205021488, 51293760, 3352152, 29514, 1;

Row n lists the coefficients of P(n,x) in descending powers of x, where P(n,x) = (1+2*n*x)*P(n-1,x) + (2*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*x^2)*D[P[n - 1, x], x]];
Table[Reverse[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 + 2*n*x)*v[i - 1] + (2*x - 2*x^2)*deriv(v[i - 1], x)); v};
row(n, p) = {my(c = Vecrev(p)); c = concat(c, vector(n + 1 - #c)); c = Vecrev(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 >= 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-e2b66c2a000699df
• 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-e2b66c2a000699df
• 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-e2b66c2a000699df
• Interlacing of consecutive rows. Rows P_(m-1), P_m interlace (weak, previous interlaces next) for every m >= 1
  The roots of consecutive polynomials alternate on the real line.
  by the cited steps: e-e2b66c2a000699df

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-e2b66c2a000699df root: theorem check

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

Used in 3 proof steps

• e-261fc200d1c6c7f8 obreschkoff_combination → obreschkoff_combination of A decomposition of the double factorials A001147.
• e-261fc200d1c6c7f8 obreschkoff_combination → obreschkoff_combination of A decomposition of the double factorials A001147.
• e-5c085bd25a6bcbdf preserver_image (reversal, shift 0) → A triangular decomposition of the double factorial numbers A001147.

Related sequences ?

Same fingerprint (1)

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

• A185411: A triangular decomposition of the double factorial numbers A001147.

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