basis-map coefficients of A021012 → basis A013610

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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:
   1:       1;
   2:      -1,       -3;
   3:       2,       12,         18;
   4:      -6,      -54,       -162,       -162;
   5:      24,      288,       1296,       2592,        1944;
   6:    -120,    -1800,     -10800,     -32400,      -48600,       -29160;
   7:     720,    12960,      97200,     388800,      874800,      1049760,       524880;
   8:   -5040,  -105840,    -952560,   -4762800,   -14288400,    -25719120,    -25719120,    -11022480;
   9:   40320,   967680,   10160640,   60963840,   228614400,    548674560,    823011840,    705438720,    264539520;
  10: -362880, -9797760, -117573120, -823011840, -3703553280, -11110659840, -22221319680, -28570268160, -21427701120, -7142567040;

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

P[1, x_] := 1;
P[n_, x_] := P[n, x] = Expand[-(n - 1 + (3*n - 3)*x)*P[n - 1, x]];
Table[CoefficientList[P[n, x], x], {n, 1, 10}] // Flatten

P(N) = {my(v = vector(N)); v[1] = 1; for(n = 2, N, my(i = n); v[i] = -(n - 1 + (3*n - 3)*x)*v[i - 1]); v};
row(n, p) = {my(c = Vecrev(p)); 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 >= 1, 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-46ba877ad14f98f3
• Coefficients of one sign. Every nonzero row, n >= 1: 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-46ba877ad14f98f3
• Hadamard kernel (Malo). Rows P_n, n >= 1, 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-46ba877ad14f98f3

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-46ba877ad14f98f3

  Geometric basis map (elementary) polytool ?

  • premise (sequence): basis-map coefficients of Triangle of coefficients in expansion of x^n in terms of Laguerre polynomials L_n(x).
    uses: Every row P_n, n >= 1, is zero or real-rooted with all roots in (-inf, 0]
    • parent_provers root_domain e-51f2fe49bf45c5d7 root: theorem check

      Parent-row provers of polytool polytool ?

    • preserver_image mobius, shift 0 e-bb5cb95fed86aa9c

      Affine, reversal and Möbius substitutions (elementary) polytool ?

      • premise (sequence): basis-map coefficients of Triangle of coefficients in expansion of x^n in terms of Laguerre polynomials L_n(x).
        uses: Every row P_n, n >= 1, is zero or real-rooted with all roots in (-inf, 0]
        • ↵ parent_provers root_domain e-51f2fe49bf45c5d7 (expanded above)

Related sequences ?

Same fingerprint (16)

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

• A013610: Triangle of coefficients in expansion of (1+3*x)^n.
• A120910 → basis A007318
• A038279 → basis A013610
• basis-map coefficients of A026729 → basis A013610
• Triangle whose (i,j)-th entry is binomial(i,j)*12^(i-j)*4^j.
• A130123 → basis A013610
• A013610 → D
• A136572 → basis A013610
• A027465: Cube of lower triangular normalized binomial matrix.
• basis-map coefficients of A112227 → basis A013610
• A120909 → x ↦ x/(x+1)
• basis-map coefficients of A030528, shifted → basis A013610
• A027465 → D
• A038243 → basis A013609
• basis-map coefficients of A026729, shifted → basis A013610
• A013610 → basis A013611