A135838 → basis A008277

OEIS matches ?

No OEIS matches.

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:     1;
  1:     4,      2;
  2:     8,     10,       2;
  3:    32,     76,      36,       4;
  4:    64,    260,     220,      56,       4;
  5:   256,   1688,    2280,    1000,     160,       8;
  6:   512,   5320,   10808,    7280,    1960,     216,      8;
  7:  2048,  32944,   97104,   94416,   38080,    6944,    560,    16;
  8:  4096, 100880,  421360,  569184,  322896,   86688,  11424,   704,   16;
  9: 16384, 613472, 3572640, 6534560, 5013120, 1859424, 356160, 35520, 1728, 32;

The row polynomials satisfy P(n,x) = (8+10*x+2*x^2)*P(n-2,x) + (10*x+4*x^2)*d/dx P(n-2,x) + 2*x^2*d^2/dx^2 P(n-2,x) for n >= 2, with P(0,x) = 1, P(1,x) = 4 + 2*x.

P[0, x_] := 1;
P[1, x_] := 4 + 2*x;
P[n_, x_] := P[n, x] = Expand[(8 + 10*x + 2*x^2)*P[n - 2, x] + (10*x + 4*x^2)*D[P[n - 2, x], x] + 2*x^2*D[P[n - 2, x], {x, 2}]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten

P(N) = {my(v = vector(N + 1)); v[1] = 1; v[2] = 4 + 2*x; for(n = 2, N, my(i = n + 1); v[i] = (8 + 10*x + 2*x^2)*v[i - 2] + (10*x + 4*x^2)*deriv(v[i - 2], x) + 2*x^2*deriv(deriv(v[i - 2], x), x)); 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-f15543b521e0834d
• 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-f15543b521e0834d
• 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-f15543b521e0834d

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-f15543b521e0834d

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

  • premise (sequence): Triangle read by rows: T(n,k) = 2^floor(n/2)*binomial(n-1,k-1).
    uses: Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, 0]
    • preserver_image reversal, shift 0 e-07f7d31230240b98

      Affine, reversal and Möbius substitutions (elementary) polytool ?

      • premise (sequence): Triangle read by rows: T(n,k) = 2^floor(n/2)*binomial(n-1,k-1).
        uses: Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, 0]
        • parent_provers periodic_product e-51cb02553cf4a8ac root: theorem check

          Parent-row provers of polytool polytool ?

    • ↵ parent_provers periodic_product e-51cb02553cf4a8ac (expanded above)

Related sequences ?

Same fingerprint (19)

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

• A178820 → basis A008277
• A344565 → basis A008277
• A126351: Triangle read by rows: matrix product of the Stirling numbers of the second kind with the binomial coefficient…
• A085881 → basis A008277
• A196347 → basis A008277
• A143494: Triangle read by rows: 2-Stirling numbers of the second kind.
• A038338 → basis A008277
• basis-map coefficients of A113214, shifted → basis A008277
• A143494 → reverse
• A156992 → basis A008277
• A356546 → basis A008277
• A028326 → basis A008277
• A038221 → basis A008277
• A178821 → basis A008277
• A178822 → basis A008277
• A038208 → basis A008277
• A038325 → basis A008277
• A085880 → basis A008277
• A126351 → reverse

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