A178821 → 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:     10,        5;
  2:     60,       75,       15;
  3:    280,      665,      315,        35;
  4:   1120,     4550,     3850,       980,        70;
  5:   4032,    26586,    35910,     15750,      2520,      126;
  6:  13440,   139650,   283710,    191100,     51450,     5670,     210;
  7:  42240,   679470,  2002770,   1947330,    785400,   143220,   11550,    330;
  8: 126720,  3120975, 13035825,  17609130,   9989595,  2681910,  353430,  21780,   495;
  9: 366080, 13707265, 79826175, 146006575, 112011900, 41546505, 7957950, 793650, 38610, 715;

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

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

P(N) = {my(v = vector(N + 1)); v[1] = 1; for(n = 1, N, my(i = n + 1); v[i] = ((2*n + 8 + (n + 4)*x)*v[i - 1] + (n + 4)*x*deriv(v[i - 1], x))/(n)); 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-2482ec2370e1f9d0
• 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-2482ec2370e1f9d0
• 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-2482ec2370e1f9d0

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-2482ec2370e1f9d0

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

  • premise (sequence): Triangle read by rows: T(n,k) = binomial(n+4,4) * binomial(n,k), 0 <= k <= n.
    uses: Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, 0]
    • theorem_check liu_wang e-20c79d8930f25d5b root: theorem check

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

    • preserver_image reversal, shift 0 e-12f9ed6b8198f9ce

      Affine, reversal and Möbius substitutions (elementary) polytool ?

      • premise (sequence): Triangle read by rows: T(n,k) = binomial(n+4,4) * binomial(n,k), 0 <= k <= n.
        uses: Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, 0]
        • ↵ theorem_check liu_wang e-20c79d8930f25d5b (expanded above)

Related sequences ?

Same fingerprint (19)

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

• A143494 → reverse
• A196347 → basis A008277
• A356546 → basis A008277
• A178822 → basis A008277
• A085881 → basis A008277
• A126351 → reverse
• A085880 → basis A008277
• A156992 → basis A008277
• A038338 → basis A008277
• A143494: Triangle read by rows: 2-Stirling numbers of the second kind.
• A135838 → basis A008277
• A126351: Triangle read by rows: matrix product of the Stirling numbers of the second kind with the binomial coefficient…
• A178820 → basis A008277
• A038221 → basis A008277
• basis-map coefficients of A113214, shifted → basis A008277
• A038208 → basis A008277
• A038325 → basis A008277
• A028326 → basis A008277
• A344565 → 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