A085880 → 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:
   1:       1;
   2:       2,        1;
   3:       8,       10,         2;
   4:      40,       95,        45,         5;
   5:     224,      910,       770,       196,        14;
   6:    1344,     8862,     11970,      5250,       840,        42;
   7:    8448,    87780,    178332,    120120,     32340,      3564,      132;
   8:   54912,   883311,   2603601,   2531529,   1021020,    186186,    15015,     429;
   9:  366080,  9016150,  37659050,  50870820,  28858830,   7747740,  1021020,   62920,   1430;
  10: 2489344, 93209402, 542817990, 992844710, 761680920, 282516234, 54114060, 5396820, 262548, 4862;

The row polynomials satisfy n*P(n,x) = (8*n-12+(4*n-6)*x)*P(n-1,x) + (4*n-6)*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[Together[((8*n - 12 + (4*n - 6)*x)*P[n - 1, x] + (4*n - 6)*x*D[P[n - 1, x], x])/(n)]];
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] = ((8*n - 12 + (4*n - 6)*x)*v[i - 1] + (4*n - 6)*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, 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-63a786f336006cd7
• 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-63a786f336006cd7
• 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-63a786f336006cd7

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-63a786f336006cd7

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

  • premise (sequence): Triangle T(n,k) read by rows: multiply row n of Pascal's triangle (A007318) by the n-th Catalan number (A000108).
    uses: Every row P_n, n >= 1, is zero or real-rooted with all roots in (-inf, 0]
    • theorem_check liu_wang e-68a5cf53bdad0f8f root: theorem check

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

    • preserver_image reversal, shift 0 e-93440eb5a87a1e1b

      Affine, reversal and Möbius substitutions (elementary) polytool ?

      • premise (sequence): Triangle T(n,k) read by rows: multiply row n of Pascal's triangle (A007318) by the n-th Catalan number (A000108).
        uses: Every row P_n, n >= 1, is zero or real-rooted with all roots in (-inf, 0]
        • ↵ theorem_check liu_wang e-68a5cf53bdad0f8f (expanded above)

Related sequences ?

Same fingerprint (19)

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

• A038221 → basis A008277
• A156992 → basis A008277
• A143494: Triangle read by rows: 2-Stirling numbers of the second kind.
• A038325 → basis A008277
• A028326 → basis A008277
• A085881 → basis A008277
• A196347 → basis A008277
• basis-map coefficients of A113214, shifted → basis A008277
• A178822 → basis A008277
• A178821 → basis A008277
• A344565 → basis A008277
• A038338 → basis A008277
• A126351: Triangle read by rows: matrix product of the Stirling numbers of the second kind with the binomial coefficient…
• A143494 → reverse
• A038208 → basis A008277
• A126351 → reverse
• A135838 → basis A008277
• A356546 → basis A008277
• A178820 → 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