A156992 → basis A013610

OEIS matches ?

• A034325: row sums, 15 terms agree OEIS nonn (row-sum match: finite agreement only) · view at the OEIS · generated b-file 200 terms from n = 1 ?

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:          4,           6;
  2:         24,          72,           54;
  3:        192,         864,         1296,          648;
  4:       1920,       11520,        25920,        25920,          9720;
  5:      23040,      172800,       518400,       777600,        583200,        174960;
  6:     322560,     2903040,     10886400,     21772800,      24494400,      14696640,       3674160;
  7:    5160960,    54190080,    243855360,    609638400,     914457600,     823011840,     411505920,      88179840;
  8:   92897280,  1114767360,   5852528640,  17557585920,   32920473600,   39504568320,   29628426240,   12697896960,   2380855680;
  9: 1857945600, 25082265600, 150493593600, 526727577600, 1185137049600, 1777705574400, 1777705574400, 1142810726400, 428554022400, 71425670400;

The row polynomials satisfy P(n,x) = (2*n+2+(3*n+3)*x)*P(n-1,x) for n >= 1, with P(0,x) = 1.

P[0, x_] := 1;
P[n_, x_] := P[n, x] = Expand[(2*n + 2 + (3*n + 3)*x)*P[n - 1, x]];
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 + 2 + (3*n + 3)*x)*v[i - 1]); 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-d66805981c716c2b
• 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-d66805981c716c2b
• 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-d66805981c716c2b

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-d66805981c716c2b

  Geometric basis map (elementary) polytool ?

  • premise (sequence): Triangle T(n,k) = n!*binomial(n-1, k-1) for 1 <= k <= n, read by rows.
    uses: Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, 0]
    • theorem_check liu_wang e-d07da0cb5a6ac36b root: theorem check

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

    • parent_provers root_domain e-e811f0fabc9e8e88 root: theorem check

      Parent-row provers of polytool polytool ?

Related sequences ?

Same fingerprint (23)

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

• A344565 → basis A013610
• A013620 → D
• A038243 → basis A013611
• A038221 → basis A013610
• basis-map coefficients of A113214, shifted → basis A013610
• A028326 → basis A013610
• A085881 → basis A013610
• A038220: Triangle whose (i,j)-th entry is binomial(i,j)*3^(i-j)*2^j.
• A085880 → basis A013610
• A120910 → x ↦ x/(x+1)
• A120910 → basis A013609
• A038325 → basis A013610
• Triangle whose (i,j)-th entry is binomial(i,j)*12^(i-j)*8^j.
• A038208 → basis A013610
• A356546 → basis A013610
• A196347 → basis A013610
• A013620: Triangle of coefficients in expansion of (2+3x)^n.
• A038220 → D
• A178822 → basis A013610
• A038338 → basis A013610
• A135838 → basis A013610
• A178821 → basis A013610
• A178820 → basis A013610

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