A344565 → basis A007318

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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:       3;
  1:      30,      15;
  2:     180,     180,      45;
  3:     840,    1260,     630,      105;
  4:    3360,    6720,    5040,     1680,     210;
  5:   12096,   30240,   30240,    15120,    3780,     378;
  6:   40320,  120960,  151200,   100800,   37800,    7560,     630;
  7:  126720,  443520,  665280,   554400,  277200,   83160,   13860,    990;
  8:  380160, 1520640, 2661120,  2661120, 1663200,  665280,  166320,  23760,  1485;
  9: 1098240, 4942080, 9884160, 11531520, 8648640, 4324320, 1441440, 308880, 38610, 2145;

The row polynomials satisfy n*P(n,x) = (2*n+8+(n+4)*x)*P(n-1,x) for n >= 1, with P(0,x) = 3.

P[0, x_] := 3;
P[n_, x_] := P[n, x] = Expand[Together[((2*n + 8 + (n + 4)*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] = 3; for(n = 1, N, my(i = n + 1); v[i] = ((2*n + 8 + (n + 4)*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-5d53734a383a65e0
• 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-5d53734a383a65e0, e-ecb3a914f88de7f3
• 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-5d53734a383a65e0, e-ecb3a914f88de7f3
• 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-5d53734a383a65e0, e-ecb3a914f88de7f3
• 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-ecb3a914f88de7f3

Proof chains

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

• basis_map A007318 e-5d53734a383a65e0

  Geometric basis map (elementary) polytool ?

  • premise (sequence): Triangle read by rows, for 0 <= k <= n: T(n, k) = binomial(n, k) * binomial(binomial(n + 3, 2), 2).
    uses: Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, 0]
    • theorem_check liu_wang e-e2cde2e22920ce64 root: theorem check

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

    • preserver_image reversal, shift 0 e-4dae17a76888d380

      Affine, reversal and Möbius substitutions (elementary) polytool ?

      • premise (sequence): Triangle read by rows, for 0 <= k <= n: T(n, k) = binomial(n, k) * binomial(binomial(n + 3, 2), 2).
        uses: Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, 0]
        • ↵ theorem_check liu_wang e-e2cde2e22920ce64 (expanded above)
• preserver_image affine, shift 0 e-ecb3a914f88de7f3

  Affine, reversal and Möbius substitutions (elementary) polytool ?

  • premise (sequence): Triangle read by rows, for 0 <= k <= n: T(n, k) = binomial(n, k) * binomial(binomial(n + 3, 2), 2).
    uses: Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, 0]
    • ↵ theorem_check liu_wang e-e2cde2e22920ce64 (expanded above)
    • ↵ preserver_image reversal, shift 0 e-4dae17a76888d380 (expanded above)

Related sequences ?

Same fingerprint (67)

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

• A085880 → x ↦ x+1
• A085880 → x ↦ x/(x+1)
• A013609 → D
• basis-map coefficients of A026729 → basis A013609
• A028326 → basis A013611
• A156992 → basis A007318
• basis-map coefficients of A026729, shifted → basis A013609
• A027466 → basis A013611
• A178820 → basis A007318
• A038208 → basis A007318
• A007318 → ×(1+(3/2)n+(1/2)n²) → x ↦ x/(x+1)
• A038208 → x ↦ x/(x+1)
• A038338 → basis A013611
• A178821 → x ↦ x/(x+1)
• A178821 → basis A013611
• A135838 → x ↦ x+1
• A007318 → D → x ↦ x/(x+1)
• A038221 → basis A013611
• A135838 → x ↦ x/(x+1)
• A178820 → x ↦ x/(x+1)
• A159854 → x ↦ x/(x+1)
• A013609 → basis A013610
• A007318 → ×(−(1/2)n+(1/2)n²) → x ↦ x+1
• A156992 → basis A013611
• A038208 → basis A013611
• A013609: Triangle of coefficients in expansion of (1+2*x)^n.
• A178822 → basis A013611
• A120909 → basis A013610
• A007318 → ×(−(1/2)n+(1/2)n²) → x ↦ x/(x+1)
• A038338 → basis A007318
• A007318 → ×(1+2n+n²) → x ↦ x/(x+1)
• A120910 → D
• A196347 → basis A013611
• A038207: Triangle whose (i,j)-th entry is binomial(i,j)*2^(i-j).
• A120909 → D
• A136572 → basis A013609
• A356546 → basis A013611
• A028326 → x ↦ x/(x+1)
• basis-map coefficients of A112227 → basis A013609
• A007318 → ×(1+2n+n²) → x ↦ x+1
• A085880 → basis A013611
• A196347 → basis A007318
• A085881 → basis A007318
• A130123 → basis A013609
• A178820 → basis A013611
• basis-map coefficients of A113214, shifted → x ↦ x+1
• A038330 → basis A013609
• A007318 → ×(1+(3/2)n+(1/2)n²) → x ↦ x+1
• A178822 → x ↦ x/(x+1)
• A344565 → basis A013611
• A038243 → basis A013610
• A135838 → basis A013611
• A120910: Triangle read by rows: T(n,k) is the number of ternary words of length n having k levels (n >= 1, 0 <= k <= n-…
• A178822 → x ↦ x+1
• A085881 → basis A013611
• A344565 → x ↦ x/(x+1)
• A178821 → basis A007318
• A120909: Triangle read by rows: T(n,k) is the number of ternary words of length n having k runs (i.e., subwords of maxi…
• A038325 → basis A013611
• basis-map coefficients of A021012 → basis A013609
• basis-map coefficients of A113214, shifted → x ↦ x/(x+1)
• basis-map coefficients of A030528, shifted → basis A013609
• A038207 → D
• A356546 → basis A007318
• A038325 → basis A007318
• basis-map coefficients of A113214, shifted → basis A013611
• A028326 → basis A007318

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