A038336 → 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:             484,               540,                100;
  1:           10648,             22120,               9600,               1000;
  2:          234256,            814320,             624400,             148000,               10000;
  3:         5153632,          28400800,           34368000,           13940000,             2100000,             100000;
  4:       113379904,         960361920,         1727464000,         1068560000,           269600000,           28200000,            1000000;
  5:      2494357888,       31865380480,        82157107200,        72839760000,         27400800000,         4726400000,          364000000,          10000000;
  6:     54875873536,     1044635754240,      3769252307200,      4609238592000,       2427247200000,       614308800000,        77112000000,        4560000000,        100000000;
  7:   1207269217792,    33977102871040,    168754954444800,    277372929856000,     196581712320000,     68502705600000,     12466272000000,     1190640000000,      55800000000,      1000000000;
  8:  26559922791424,  1099339984051200,   7427479115392000,  16110941896960000,   14961795462400000,   6898011926400000,   1707261360000000,   234201600000000,   17598000000000,    670000000000,   10000000000;
  9: 584318301411328, 35444478717552640, 322947522686976000, 912043769795840000, 1088740737638400000, 646274813324800000, 208975550784000000, 38619160800000000, 4137012000000000, 251020000000000, 7920000000000, 100000000000;

The row polynomials satisfy P(n,x) = (22+10*x)*P(n-1,x) + 10*x*d/dx P(n-1,x) for n >= 1, with P(0,x) = 484 + 540*x + 100*x^2.

P[0, x_] := 484 + 540*x + 100*x^2;
P[n_, x_] := P[n, x] = Expand[(22 + 10*x)*P[n - 1, x] + 10*x*D[P[n - 1, x], x]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten

P(N) = {my(v = vector(N + 1)); v[1] = 484 + 540*x + 100*x^2; for(n = 1, N, my(i = n + 1); v[i] = (22 + 10*x)*v[i - 1] + 10*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 - 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-79186619a2157f5c
• 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-79186619a2157f5c
• 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-79186619a2157f5c

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-79186619a2157f5c

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

  • premise (sequence): Triangle whose (i,j)-th entry is binomial(i,j)*12^(i-j)*10^j.
    uses: Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, 0]
    • theorem_check liu_wang e-2f1e093ae220e593 root: theorem check

      Liu–Wang theorem [LW07, Thm 2.1, Cor 2.4] ?

    • parent_provers periodic_product e-3f0a5f7ceded39cb root: theorem check

      Parent-row provers of polytool polytool ?

Related sequences ?

Same fingerprint (1)

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

• A038259 → basis A008277

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
2. [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