A038284 → basis A013611

OEIS matches ?

• A009982: row sums, 15 terms agree OEIS nonn (row-sum match: finite agreement only) · view at the OEIS

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:           196,            672,             576;
  1:          2744,          14112,           24192,            13824;
  2:         38416,         263424,          677376,           774144,            331776;
  3:        537824,        4609920,        15805440,         27095040,          23224320,           7962624;
  4:       7529536,       77446656,       331914240,        758661120,         975421440,         668860416,         191102976;
  5:     105413504,     1264962048,      6505519104,      18587197440,       31863767040,       32774160384,       18728091648,        4586471424;
  6:    1475789056,    20239392768,    121436356608,     416353222656,      892185477120,     1223568654336,     1048773132288,      513684799488,      110075314176;
  7:   20661046784,   318770436096,   2185854418944,    8743417675776,    22483074023424,    38542412611584,    44048471556096,    32362142367744,    13869489586176,     2641807540224;
  8:  289254654976,  4958651228160,  38252452331520,  174868353515520,   524605060546560,  1079187553124352,  1541696504463360,  1510233310494720,   970864271032320,   369853055631360,   63403380965376;
  9: 4049565169664, 76363228913664, 654541962117120, 3366215805173760, 11541311332024320, 27699147196858368, 47484252337471488, 58143982454046720, 49837699246325760, 28478685283614720, 9764120668667904, 1521681143169024;

The row polynomials satisfy P(n,x) = (14+24*x)*P(n-1,x) for n >= 1, with P(0,x) = 196 + 672*x + 576*x^2.

P[0, x_] := 196 + 672*x + 576*x^2;
P[n_, x_] := P[n, x] = Expand[(14 + 24*x)*P[n - 1, x]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten

P(N) = {my(v = vector(N + 1)); v[1] = 196 + 672*x + 576*x^2; for(n = 1, N, my(i = n + 1); v[i] = (14 + 24*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-06104b6211fa5aaa
• 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-06104b6211fa5aaa
• 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-06104b6211fa5aaa

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 A013611 e-06104b6211fa5aaa

  Geometric basis map (elementary) polytool ?

  • premise (sequence): Triangle whose (i,j)-th entry is binomial(i,j)*8^(i-j)*6^j.
    uses: Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, 0]
    • theorem_check liu_wang e-4ef268212a4f2f46 root: theorem check

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

    • parent_provers periodic_product e-4fdd1b3030c1c25c root: theorem check

      Parent-row provers of polytool polytool ?

Related sequences ?

Same fingerprint (3)

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

• Triangle whose (i,j)-th entry is binomial(i,j)*12^(i-j)*7^j.
• A038335 → basis A013611
• A013613 → basis A013609

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