A200536: Triangle, read by rows of 2*n+1 terms, where row n lists the coefficients in (1+3*x+2*x^2)^n.

OEIS matches ?

• A000400: 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 ?
• A200536: triangle, 64 terms agree, 8 full rows row lengths from the atlas (tabf) OEIS nonn in polytool catalog · view at the OEIS · generated b-file 1024 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;
  1: 1,  3,   2;
  2: 1,  6,  13,   12,     4;
  3: 1,  9,  33,   63,    66,    36,      8;
  4: 1, 12,  62,  180,   321,   360,    248,     96,     16;
  5: 1, 15, 100,  390,   985,  1683,   1970,   1560,    800,     240,      32;
  6: 1, 18, 147,  720,  2355,  5418,   8989,  10836,   9420,    5760,    2352,     576,      64;
  7: 1, 21, 203, 1197,  4809, 13923,  29953,  48639,  59906,   55692,   38472,   19152,    6496,   1344,    128;
  8: 1, 24, 268, 1848,  8806, 30744,  81340, 166344, 265729,  332688,  325360,  245952,  140896,  59136,  17152,   3072,   256;
  9: 1, 27, 342, 2700, 14886, 60858, 191184, 471852, 927441, 1462563, 1854882, 1887408, 1529472, 973728, 476352, 172800, 43776, 6912, 512;

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

P[0, x_] := 1;
P[n_, x_] := P[n, x] = Expand[(1 + 3*x + 2*x^2)*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] = (1 + 3*x + 2*x^2)*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:
  0: 1;
  1: 1,  3,   2;
  2: 1,  6,  13,   12,     4;
  3: 1,  9,  33,   63,    66,    36,      8;
  4: 1, 12,  62,  180,   321,   360,    248,     96,     16;
  5: 1, 15, 100,  390,   985,  1683,   1970,   1560,    800,     240,      32;
  6: 1, 18, 147,  720,  2355,  5418,   8989,  10836,   9420,    5760,    2352,     576,      64;
  7: 1, 21, 203, 1197,  4809, 13923,  29953,  48639,  59906,   55692,   38472,   19152,    6496,   1344,    128;
  8: 1, 24, 268, 1848,  8806, 30744,  81340, 166344, 265729,  332688,  325360,  245952,  140896,  59136,  17152,   3072,   256;
  9: 1, 27, 342, 2700, 14886, 60858, 191184, 471852, 927441, 1462563, 1854882, 1887408, 1529472, 973728, 476352, 172800, 43776, 6912, 512;

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

P[0, x_] := 1;
P[n_, x_] := P[n, x] = Expand[(1 + 3*x + 2*x^2)*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] = (1 + 3*x + 2*x^2)*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-982ad459e6d62a8e
• 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-982ad459e6d62a8e
• 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-982ad459e6d62a8e

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

• parent_provers periodic_product e-982ad459e6d62a8e root: theorem check

  Parent-row provers of polytool polytool ?

Used in 5 proof steps

• e-185113cbc4b6c77b basis_map (A013609) → basis_map_A013609 of Triangle, read by rows of 2*n+1 terms, where row n lists the coeffic…
• e-4c5ee4490c6f16fe basis_map (A008277) → basis_map_A008277 of Triangle, read by rows of 2*n+1 terms, where row n lists the coeffic…
• e-6e97878f95ac8f6d basis_map (A007318) → basis_map_A007318 of Triangle, read by rows of 2*n+1 terms, where row n lists the coeffic…
• e-bf76d3591eaf645b basis_map (A013610) → basis_map_A013610 of Triangle, read by rows of 2*n+1 terms, where row n lists the coeffic…
• e-ecef41f482d88baf basis_map (A013611) → basis_map_A013611 of Triangle, read by rows of 2*n+1 terms, where row n lists the coeffic…

Related sequences ?

No candidate equalities or related sequences.