A344565 → basis A013609

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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,      30;
  2:     180,     360,      180;
  3:     840,    2520,     2520,      840;
  4:    3360,   13440,    20160,    13440,      3360;
  5:   12096,   60480,   120960,   120960,     60480,     12096;
  6:   40320,  241920,   604800,   806400,    604800,    241920,    40320;
  7:  126720,  887040,  2661120,  4435200,   4435200,   2661120,   887040,   126720;
  8:  380160, 3041280, 10644480, 21288960,  26611200,  21288960, 10644480,  3041280,  380160;
  9: 1098240, 9884160, 39536640, 92252160, 138378240, 138378240, 92252160, 39536640, 9884160, 1098240;

The row polynomials satisfy n*P(n,x) = (2*n+8+(2*n+8)*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 + (2*n + 8)*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 + (2*n + 8)*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-47e5b69025eff315
• 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-47e5b69025eff315
• 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-47e5b69025eff315
• 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-47e5b69025eff315

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 A013609 e-47e5b69025eff315

  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)

Related sequences ?

Same fingerprint (59)

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

• A094305: Triangle read by rows: T(n,k) = ((n+1)(n+2)/2) * binomial(n,k) (0 <= k <= n).
• A085881: Triangle T(n,k) read by rows: multiply row n of Pascal's triangle (A007318) by A001147(n).
• A178822 → D
• A135838: Triangle read by rows: T(n,k) = 2^floor(n/2)*binomial(n-1,k-1).
• A196347 → basis A013609
• A038208: Triangle whose (i,j)-th entry is binomial(i,j)*2^i.
• A129533: Array read by antidiagonals: T(n,k) = binomial(n+1,2)*binomial(n+k,n+1) for 0 <= k <= n.
• A003506: Triangle of denominators in Leibniz's Harmonic Triangle a(n,k), n >= 1, 1 <= k <= n.
• A038221: Triangle whose (i,j)-th entry is binomial(i,j)*3^(i-j)*3^j.
• Triangle whose (i,j)-th entry is binomial(i,j)*12^(i-j)*12^j.
• A237765: Triangular array read by rows: T(n,k) = binomial(n,2)*binomial(n,k), n>=0, 0<=k<=n.
• A178822 → basis A013609
• A356546 → basis A013609
• Riordan array (1-x,x/(1-x)).
• A178821 → D
• A135065: A127733 * A007318 as infinite lower triangular matrices.
• basis-map coefficients of A113214, shifted → basis A013609
• A135838 → D
• A038325: Triangle read by rows whose (i,j)-th entry is binomial(i,j)*11^(i-j)*11^j.
• A010054 → reverse → x ↦ x+1
• A007318 → ×(−(1/2)n+(1/2)n²) → D
• A038221 → basis A013609
• basis-map coefficients of Riordan array (1+2x,x(1+x)). shifted by x -> -x + 0
• A178820 → D
• A027465 → basis A013611
• A344565 → D
• basis-map coefficients of A112227 → x ↦ x+1
• A120910 → basis A013610
• A178820 → basis A013609
• A178821 → basis A013609
• basis-map coefficients of A026729, shifted → basis A007318
• A085880 → D
• A085881 → basis A013609
• A038325 → basis A013609
• basis-map coefficients of A113214, shifted → D
• A356546: Triangle read by rows. T(n, k) = RisingFactorial(n + 1, n) / (k! * (n - k)!).
• A085880 → basis A013609
• A038208 → D
• A178820: Triangle read by rows: T(n,k) = C(n+3,3) * C(n,k), 0 <= k <= n.
• A038338 → basis A013609
• basis-map coefficients of A021012 → x ↦ x+1
• A344565: Triangle read by rows, for 0 <= k <= n: T(n, k) = binomial(n, k) * binomial(binomial(n + 3, 2), 2).
• A007318: Binomial
• A085880: Triangle T(n,k) read by rows: multiply row n of Pascal's triangle (A007318) by the n-th Catalan number (A00010…
• basis-map coefficients of A026729 → basis A007318
• A178821: Triangle read by rows: T(n,k) = binomial(n+4,4) * binomial(n,k), 0 <= k <= n.
• A156992 → basis A013609
• A007318 → ×(1+2n+n²) → D
• A007318 → ×(1+(3/2)n+(1/2)n²) → D
• A216973: Exponential Riordan array [x*exp(x),x].
• A028326 → basis A013609
• A178822: Triangle read by rows: T(n,k) = C(n+5,5) * C(n,k), 0 <= k <= n.
• A007318 → D → D
• A156992: Triangle T(n,k) = n!*binomial(n-1, k-1) for 1 <= k <= n, read by rows.
• A196347: Triangle T(n, k) read by rows, T(n, k) = n!*binomial(n, k).
• A135838 → basis A013609
• A038330 → basis A013611
• A028326 → D
• A028326: Twice Pascal's triangle A007318: T(n,k) = 2*C(n,k).

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