A178822 → basis A013610

OEIS matches ?

• A050982: row sums, 15 terms agree OEIS nonn (row-sum match: finite agreement only) · view at the OEIS · generated b-file 200 terms from n = 5 ?

A generated b-file is computed from the node's definition along the row alignment of its OEIS link and starts at the entry's offset. It is an identification on a finite prefix, not a proof that the node and the OEIS entry agree for every n.

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:      12,       18;
  2:      84,      252,      189;
  3:     448,     2016,     3024,      1512;
  4:    2016,    12096,    27216,     27216,     10206;
  5:    8064,    60480,   181440,    272160,    204120,     61236;
  6:   29568,   266112,   997920,   1995840,   2245320,   1347192,    336798;
  7:  101376,  1064448,  4790016,  11975040,  17962560,  16166304,   8083152,   1732104;
  8:  329472,  3953664, 20756736,  62270208, 116756640, 140107968, 105080976,  45034704,   8444007;
  9: 1025024, 13837824, 83026944, 290594304, 653837184, 980755776, 980755776, 630485856, 236432196, 39405366;

The row polynomials satisfy n*P(n,x) = (2*n+10+(3*n+15)*x)*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 + 10 + (3*n + 15)*x)*P[n - 1, 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 + 10 + (3*n + 15)*x)*v[i - 1])/(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-97dd7dd90f768ab3
• 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-97dd7dd90f768ab3
• 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-97dd7dd90f768ab3
• Interlacing of consecutive rows. Rows P_(m-1), P_m interlace (weak, unordered) for every m >= 1
  The roots of consecutive polynomials alternate on the real line.
  by the cited steps: e-97dd7dd90f768ab3

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 A013610 e-97dd7dd90f768ab3

  Geometric basis map (elementary) polytool ?

  • premise (sequence): Triangle read by rows: T(n,k) = C(n+5,5) * C(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-548a05679c99d179 root: theorem check

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

    • preserver_image reversal, shift 0 e-0d5dad6bd99a01a4

      Affine, reversal and Möbius substitutions (elementary) polytool ?

      • premise (sequence): Triangle read by rows: T(n,k) = C(n+5,5) * C(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-548a05679c99d179 (expanded above)

Related sequences ?

Same fingerprint (23)

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

• A356546 → basis A013610
• Triangle whose (i,j)-th entry is binomial(i,j)*12^(i-j)*8^j.
• A013620: Triangle of coefficients in expansion of (2+3x)^n.
• A178820 → basis A013610
• A038338 → basis A013610
• A038325 → basis A013610
• A135838 → basis A013610
• A120910 → basis A013609
• A344565 → basis A013610
• A196347 → basis A013610
• A038208 → basis A013610
• A038220: Triangle whose (i,j)-th entry is binomial(i,j)*3^(i-j)*2^j.
• A085880 → basis A013610
• A028326 → basis A013610
• A038220 → D
• basis-map coefficients of A113214, shifted → basis A013610
• A156992 → basis A013610
• A013620 → D
• A038221 → basis A013610
• A178821 → basis A013610
• A085881 → basis A013610
• A038243 → basis A013611
• A120910 → x ↦ x/(x+1)

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