basis-map coefficients of A021012 → basis A008277

OEIS matches ?

• A320962: row sums, 15 terms agree (row-sum match: finite agreement only) · view at the OEIS

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:
   1:       1;
   2:      -1,         -1;
   3:       2,          6,           2;
   4:      -6,        -42,         -36,           -6;
   5:      24,        360,         600,          240,           24;
   6:    -120,      -3720,      -10800,        -7800,        -1800,        -120;
   7:     720,      45360,      216720,       252000,       100800,       15120,         720;
   8:   -5040,    -640080,    -4868640,     -8573040,     -5292000,    -1340640,     -141120,      -5040;
   9:   40320,   10281600,   121968000,    313286400,    280264320,   106686720,    18627840,    1451520,     40320;
  10: -362880, -185431680, -3385670400, -12376022400, -15431472000, -8283461760, -2133734400, -272160000, -16329600, -362880;

The row polynomials satisfy P(n,x) = -(n-1+(n-1)*x)*P(n-1,x) - (n-1)*x*d/dx 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 + (n - 1)*x)*P[n - 1, x] - (n - 1)*x*D[P[n - 1, x], 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 + (n - 1)*x)*v[i - 1] - (n - 1)*x*deriv(v[i - 1], x)); 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-f4658f48c67ff3dd
• 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-f4658f48c67ff3dd
• 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-f4658f48c67ff3dd

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 A008277 e-f4658f48c67ff3dd

  Ray basis map theorem [MW08, Thm 2] polytool RealRooted rayBasisMap_maps_realRootedIn_iff ?

  • 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 (11)

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

• basis-map coefficients of A030528, shifted → basis A008277
• basis-map coefficients of A112227 → basis A008277
• A008277: Triangle of Stirling numbers of the second kind, S2(n,k), n >= 1, 1 <= k <= n.
• A008278: Reflected triangle of Stirling numbers of 2nd kind, S(n,n-k+1), n >= 1, 1 <= k <= n.
• basis-map coefficients of A026729, shifted → basis A008277
• A106800: Triangle of Stirling numbers of 2nd kind, S(n, n-k), n >= 0, 0 <= k <= n.
• A130123 → basis A008277
• A048993: Triangle of Stirling numbers of 2nd kind, S(n,k), n >= 0, 0 <= k <= n.
• A106800 → reverse (shift +1) → reverse
• A023531 → basis A008277
• A136572 → basis A008277

References

1. [MW08] Shi-Mei Ma and Yi Wang. q-Eulerian polynomials and polynomials with only real zeros. The Electronic Journal of Combinatorics, 15(1):Research Paper 17, January 2008. doi:10.37236/741