A356546 → basis A013610

OEIS matches ?

No OEIS matches.

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:        4,         6;
  2:       24,        72,         54;
  3:      160,       720,       1080,        540;
  4:     1120,      6720,      15120,      15120,        5670;
  5:     8064,     60480,     181440,     272160,      204120,       61236;
  6:    59136,    532224,    1995840,    3991680,     4490640,     2694384,      673596;
  7:   439296,   4612608,   20756736,   51891840,    77837760,    70053984,    35026992,     7505784;
  8:  3294720,  39536640,  207567360,  622702080,  1167566400,  1401079680,  1050809760,   450347040,   84440070;
  9: 24893440, 336061440, 2016368640, 7057290240, 15878903040, 23818354560, 23818354560, 15311799360, 5741924760, 956987460;

The row polynomials satisfy n*P(n,x) = (8*n-4+(12*n-6)*x)*P(n-1,x) for n >= 1, with P(0,x) = 1.

P[0, x_] := 1;
P[n_, x_] := P[n, x] = Expand[Together[((8*n - 4 + (12*n - 6)*x)*P[n - 1, x])/(n)]];
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] = ((8*n - 4 + (12*n - 6)*x)*v[i - 1])/(n)); 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)

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-367cc697b56efa4f
• 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-367cc697b56efa4f
• 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-367cc697b56efa4f
• 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-367cc697b56efa4f

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

• basis_map A013610 e-367cc697b56efa4f

  Geometric basis map (elementary) polytool ?

  • premise (sequence): Triangle read by rows. T(n, k) = RisingFactorial(n + 1, n) / (k! * (n - k)!).
    uses: Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, 0]
    • theorem_check liu_wang e-883c9c594ff729de root: theorem check

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

Related sequences ?

Same fingerprint (23)

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

• A178822 → basis A013610
• A038220 → D
• A120910 → basis A013609
• A085880 → basis A013610
• A085881 → basis A013610
• A120910 → x ↦ x/(x+1)
• A038325 → basis A013610
• A013620: Triangle of coefficients in expansion of (2+3x)^n.
• A038208 → basis A013610
• A028326 → basis A013610
• A038221 → basis A013610
• A344565 → basis A013610
• A196347 → basis A013610
• A013620 → D
• A156992 → basis A013610
• A178820 → basis A013610
• basis-map coefficients of A113214, shifted → basis A013610
• Triangle whose (i,j)-th entry is binomial(i,j)*12^(i-j)*8^j.
• A178821 → basis A013610
• A038220: Triangle whose (i,j)-th entry is binomial(i,j)*3^(i-j)*2^j.
• A038243 → basis A013611
• A135838 → basis A013610
• A038338 → basis A013610

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