A056242: Triangle read by rows: T(n,k) = number of k-part order-consecutive partition of {1,2,...,n} (1 <= k <= n).

OEIS matches ?

• A007052: 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 ?
• A056242: triangle, 59 terms agree, 10 full rows row-aligned (tabl) OEIS nonn in polytool catalog · view at the OEIS · generated b-file 990 terms from n = 1 ?
• A048580: row sums, 14 terms agree OEIS nonn (row-sum match: finite agreement only) · view at the OEIS · generated b-file 200 terms from n = 0 ?
• A117317: reversed triangle, 58 terms agree, 10 full rows row-aligned (tabl) OEIS nonn in polytool catalog · view at the OEIS · generated b-file 990 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:
   1: 1;
   2: 1,  2;
   3: 1,  5,   4;
   4: 1,  9,  16,    8;
   5: 1, 14,  41,   44,   16;
   6: 1, 20,  85,  146,  112,    32;
   7: 1, 27, 155,  377,  456,   272,    64;
   8: 1, 35, 259,  833, 1408,  1312,   640,  128;
   9: 1, 44, 406, 1652, 3649,  4712,  3568, 1472,  256;
  10: 1, 54, 606, 3024, 8361, 14002, 14608, 9312, 3328, 512;

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

P[1, x_] := 1;
P[2, x_] := 1 + 2*x;
P[n_, x_] := P[n, x] = Expand[(2 + 2*x)*P[n - 1, x] - (1 + x)*P[n - 2, x]];
Table[CoefficientList[P[n, x], x], {n, 1, 10}] // Flatten

P(N) = {my(v = vector(N)); v[1] = 1; v[2] = 1 + 2*x; for(n = 3, N, my(i = n); v[i] = (2 + 2*x)*v[i - 1] - (1 + x)*v[i - 2]); v};
row(n, p) = {my(c = Vecrev(p)); c};
T(N) = {my(v = P(N)); concat(vector(#v, i, row(i, v[i])))};
T(10)

Triangle begins:
   1: 1;
   2: 1,  2;
   3: 1,  5,   4;
   4: 1,  9,  16,    8;
   5: 1, 14,  41,   44,   16;
   6: 1, 20,  85,  146,  112,    32;
   7: 1, 27, 155,  377,  456,   272,    64;
   8: 1, 35, 259,  833, 1408,  1312,   640,  128;
   9: 1, 44, 406, 1652, 3649,  4712,  3568, 1472,  256;
  10: 1, 54, 606, 3024, 8361, 14002, 14608, 9312, 3328, 512;

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

P[1, x_] := 1;
P[2, x_] := 1 + 2*x;
P[n_, x_] := P[n, x] = Expand[(2 + 2*x)*P[n - 1, x] - (1 + x)*P[n - 2, x]];
Table[PadRight[CoefficientList[P[n, x], x], n], {n, 1, 10}] // Flatten

P(N) = {my(v = vector(N)); v[1] = 1; v[2] = 1 + 2*x; for(n = 3, N, my(i = n); v[i] = (2 + 2*x)*v[i - 1] - (1 + x)*v[i - 2]); v};
row(n, p) = {my(c = Vecrev(p)); c = concat(c, vector(n - #c)); c};
T(N) = {my(v = P(N)); concat(vector(#v, i, row(i, v[i])))};
T(10)

Triangle begins:
  0:   1;
  1:   2,    1;
  2:   4,    5,    1;
  3:   8,   16,    9,     1;
  4:  16,   44,   41,    14,     1;
  5:  32,  112,  146,    85,    20,    1;
  6:  64,  272,  456,   377,   155,   27,    1;
  7: 128,  640, 1312,  1408,   833,  259,   35,   1;
  8: 256, 1472, 3568,  4712,  3649, 1652,  406,  44,  1;
  9: 512, 3328, 9312, 14608, 14002, 8361, 3024, 606, 54, 1;

Row n lists the coefficients of P(n,x) in descending powers of x, where P(n,x) = (2+2*x)*P(n-1,x) - (1+x)*P(n-2,x) for n >= 2, with P(0,x) = 1, P(1,x) = 1 + 2*x.

P[0, x_] := 1;
P[1, x_] := 1 + 2*x;
P[n_, x_] := P[n, x] = Expand[(2 + 2*x)*P[n - 1, x] - (1 + x)*P[n - 2, x]];
Table[Reverse[PadRight[CoefficientList[P[n, x], x], n + 1]], {n, 0, 9}] // Flatten

P(N) = {my(v = vector(N + 1)); v[1] = 1; v[2] = 1 + 2*x; for(n = 2, N, my(i = n + 1); v[i] = (2 + 2*x)*v[i - 1] - (1 + x)*v[i - 2]); v};
row(n, p) = {my(c = Vecrev(p)); c = concat(c, vector(n + 1 - #c)); c = Vecrev(c); 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 >= 1, is zero or real-rooted with all roots in [-1, 0]
  Every polynomial of the sequence has only real roots, inside the stated interval.
  by the cited steps: e-3cb5f3d64c4bb317
• Coefficients of one sign. Every nonzero row, n >= 1: 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-3cb5f3d64c4bb317
• Hadamard kernel (Malo). Rows P_n, n >= 1, 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-3cb5f3d64c4bb317

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 A056242 e-3cb5f3d64c4bb317

  Affine-current basis map theorem polytool RealRooted orderTwoConstants_nonneg_iff ?

  • premise (sequence): basis-map coefficients of Square array of binomial coefficients T(n,k) = binomial(n,k), n >= 0, k >= 0, read by downwar…
    uses: Every row P_n, n >= 1, is zero or real-rooted with all roots in [0, 0]
    no recorded property covers this requirement

Used in 9 proof steps

• e-0732c8d06b4db3af preserver_image (euler_polynomial, shift 0) → theta_plus_one of Triangle read by rows: T(n,k) = number of k-part order-consecutive part…
• e-1da1d1636f7486ed preserver_image (derivative_polynomial, shift 0) → derivative of Triangle read by rows: T(n,k) = number of k-part order-consecutive partitio…
• e-3cf80f21f053e148 preserver_image (reversal, shift 0) → reversal of Triangle read by rows: T(n,k) = number of k-part order-consecutive partition …
• e-6c5bdfbf185d1b6a preserver_image (euler_polynomial, shift 0) → theta_plus_one_squared of Triangle read by rows: T(n,k) = number of k-part order-consecut…
• e-a682124e3a957353 preserver_image (derivative_polynomial, shift 0) → one_plus_derivative of Triangle read by rows: T(n,k) = number of k-part order-consecutive…
• e-e8893cf05feee942 preserver_image (affine, shift 0) → shift_x_plus_one of Triangle read by rows: T(n,k) = number of k-part order-consecutive pa…
• e-fc6a306274ee3581 preserver_image (mobius, shift 0) → mobius_x_over_x_plus_one of Triangle read by rows: T(n,k) = number of k-part order-consec…
• e-27376e0c0e2dfb80 veronese_section (r=2, j=1, shift 0) → veronese_section_2_1 of Triangle read by rows: T(n,k) = number of k-part order-consecutiv…
• e-37a36b1b0e8be851 veronese_section (r=2, j=0, shift 0) → veronese_section_2_0 of Triangle read by rows: T(n,k) = number of k-part order-consecutiv…

Related sequences ?

Same fingerprint (1)

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

• basis-map coefficients of A026729 → basis A056242 → reverse