A391533 → reverse

OEIS matches ?

• A391533: reversed triangle, 63 terms agree, 9 full rows row lengths from the atlas (tabf) OEIS nonn in polytool catalog · view at the OEIS · generated b-file 987 terms from n = 2 ?
• A116453: row sums, 15 terms agree OEIS nonn (row-sum match: finite agreement only) · view at the OEIS
• A288135: row sums, 15 terms agree OEIS nonn (row-sum match: finite agreement only) · view at the OEIS
• A005010: row sums, 15 terms agree OEIS nonn (row-sum match: finite agreement only) · view at the OEIS · generated b-file 200 terms from n = 0 ?

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,  4,  4;
  1: 1,  5,  8,   4;
  2: 1,  6, 13,  12,   4;
  3: 1,  7, 19,  25,  16,   4;
  4: 1,  8, 26,  44,  41,  20,    4;
  5: 1,  9, 34,  70,  85,  61,   24,   4;
  6: 1, 10, 43, 104, 155, 146,   85,  28,   4;
  7: 1, 11, 53, 147, 259, 301,  231, 113,  32,   4;
  8: 1, 12, 64, 200, 406, 560,  532, 344, 145,  36,  4;
  9: 1, 13, 76, 264, 606, 966, 1092, 876, 489, 181, 40, 4;

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

P[0, x_] := 1 + 4*x + 4*x^2;
P[n_, x_] := P[n, x] = Expand[(1 + 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 + 4*x + 4*x^2; for(n = 1, N, my(i = n + 1); v[i] = (1 + 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)

Triangle begins:
   2: 4,  4,   1;
   3: 4,  8,   5,   1;
   4: 4, 12,  13,   6,   1;
   5: 4, 16,  25,  19,   7,    1;
   6: 4, 20,  41,  44,  26,    8,   1;
   7: 4, 24,  61,  85,  70,   34,   9,   1;
   8: 4, 28,  85, 146, 155,  104,  43,  10,   1;
   9: 4, 32, 113, 231, 301,  259, 147,  53,  11,  1;
  10: 4, 36, 145, 344, 532,  560, 406, 200,  64, 12,  1;
  11: 4, 40, 181, 489, 876, 1092, 966, 606, 264, 76, 13, 1;

Row n lists the coefficients of P(n,x) in descending powers of x, where P(n,x) = (1+x)*P(n-1,x) for n >= 3, with P(2,x) = 1 + 4*x + 4*x^2.

P[2, x_] := 1 + 4*x + 4*x^2;
P[n_, x_] := P[n, x] = Expand[(1 + x)*P[n - 1, x]];
Table[Reverse[CoefficientList[P[n, x], x]], {n, 2, 11}] // Flatten

P(N) = {my(v = vector(N - 1)); v[1] = 1 + 4*x + 4*x^2; for(n = 3, N, my(i = n - 1); v[i] = (1 + x)*v[i - 1]); v};
row(n, p) = {my(c = Vecrev(p)); c = Vecrev(c); c};
T(N) = {my(v = P(N)); concat(vector(#v, i, row(i + 1, v[i])))};
T(11)

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-6248136e99299279
• 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-6248136e99299279
• 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-6248136e99299279

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

• preserver_image reversal, shift 0 e-6248136e99299279

  Affine, reversal and Möbius substitutions (elementary) polytool ?

  • premise (sequence): The truncated (4,4,1)-Pascal triangle, given by 4*C(n-2, k) + 4*C(n-2, k-1) + C(n-2, k-2) for n>=2.
    uses: Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, 0]
    • theorem_check liu_wang e-3cff49c1888b5e80 root: theorem check

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

    • parent_provers periodic_product e-5e5f630c1e2baa8a root: theorem check

      Parent-row provers of polytool polytool ?

Related sequences ?

Same fingerprint (2)

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

• A391533: The truncated (4,4,1)-Pascal triangle, given by 4*C(n-2, k) + 4*C(n-2, k-1) + C(n-2, k-2) for n>=2.
• A159854 → 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