basis-map coefficients of A021012 → basis A013609

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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,       -2;
   3:       2,        8,         8;
   4:      -6,      -36,       -72,        -48;
   5:      24,      192,       576,        768,        384;
   6:    -120,    -1200,     -4800,      -9600,      -9600,       -3840;
   7:     720,     8640,     43200,     115200,     172800,      138240,       46080;
   8:   -5040,   -70560,   -423360,   -1411200,   -2822400,    -3386880,    -2257920,     -645120;
   9:   40320,   645120,   4515840,   18063360,   45158400,    72253440,    72253440,    41287680,   10321920;
  10: -362880, -6531840, -52254720, -243855360, -731566080, -1463132160, -1950842880, -1672151040, -836075520, -185794560;

The row polynomials satisfy P(n,x) = -(n-1+(2*n-2)*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 + (2*n - 2)*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 + (2*n - 2)*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-fbc81b67b4fab4f6
• 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-fbc81b67b4fab4f6
• 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-fbc81b67b4fab4f6

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-fbc81b67b4fab4f6

  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 (67)

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

• A178820 → x ↦ x/(x+1)
• A178821 → x ↦ x/(x+1)
• basis-map coefficients of A113214, shifted → x ↦ x+1
• A038208 → x ↦ x/(x+1)
• A156992 → basis A007318
• basis-map coefficients of A113214, shifted → basis A013611
• 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…
• A007318 → ×(−(1/2)n+(1/2)n²) → x ↦ x+1
• A007318 → ×(1+2n+n²) → x ↦ x/(x+1)
• A007318 → ×(1+(3/2)n+(1/2)n²) → x ↦ x/(x+1)
• A085881 → basis A013611
• A007318 → ×(1+2n+n²) → x ↦ x+1
• basis-map coefficients of A030528, shifted → basis A013609
• A135838 → basis A013611
• A344565 → x ↦ x/(x+1)
• A038338 → basis A013611
• A085880 → basis A013611
• A120910 → D
• A178821 → basis A013611
• A344565 → basis A013611
• A007318 → D → x ↦ x/(x+1)
• A085880 → x ↦ x/(x+1)
• A038330 → basis A013609
• basis-map coefficients of A026729, shifted → basis A013609
• A196347 → basis A007318
• A178822 → basis A013611
• A038208 → basis A007318
• A038208 → basis A013611
• A178822 → x ↦ x/(x+1)
• A038207 → D
• A028326 → basis A007318
• A135838 → x ↦ x+1
• A130123 → basis A013609
• A085880 → x ↦ x+1
• A159854 → x ↦ x/(x+1)
• A027466 → basis A013611
• A028326 → x ↦ x/(x+1)
• A178821 → basis A007318
• A120909 → D
• A038243 → basis A013610
• A178820 → basis A007318
• A038325 → basis A013611
• A038338 → basis A007318
• A013609 → D
• A356546 → basis A013611
• A356546 → basis A007318
• A038221 → basis A013611
• A085881 → basis A007318
• A013609: Triangle of coefficients in expansion of (1+2*x)^n.
• A038325 → basis A007318
• basis-map coefficients of A026729 → basis A013609
• A196347 → basis A013611
• A013609 → basis A013610
• A007318 → ×(−(1/2)n+(1/2)n²) → x ↦ x/(x+1)
• A120909 → basis A013610
• A178822 → x ↦ x+1
• basis-map coefficients of A113214, shifted → x ↦ x/(x+1)
• A135838 → x ↦ x/(x+1)
• A028326 → basis A013611
• basis-map coefficients of A112227 → basis A013609
• A344565 → basis A007318
• A178820 → 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-…
• A007318 → ×(1+(3/2)n+(1/2)n²) → x ↦ x+1
• A136572 → basis A013609
• A156992 → basis A013611
• A038207: Triangle whose (i,j)-th entry is binomial(i,j)*2^(i-j).