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for an input row that is a polynomial the degree condition holds automatically","lean_route":null,"lean_declarations":["Polynomial.Splits.comp_of_degree_le_one"],"certificate_hash":"abfbe5d27d628d7a5791683d17fb096fb32385fc0c2423dadf4323dec612c441","verification_polytool":"replayed","verification_lean":"not_attempted","claims":[{"claim":"real_rooted","from_index":0,"region":{"upper":{"closed":true,"value":"-1"}}}],"premises":[{"position":0,"node_id":"n-582f0cc7fb344588","role":"sequence","name":"one_plus_derivative of Triangular array T read by rows: T(n,0) = T(n,n) = 1 for all n >= 0, T(n,k) = T(n-1,k-1) + T(n-1,k) for odd n and 1< = k <= n-1, T(n,k) = T(n-1,k-1) + T(n-1,k) + T(n-2,k-1) for even n and 1 <= k <= n-1.","uses":{"claim":"real_rooted","from_index":0,"region":{"upper":{"closed":true,"value":"0"}}},"uses_text":"Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, 0]"}]}],"used_by":[],"used_by_total":0,"links":[],"oeis":[],"bfiles":[],"oeis_texts":[{"oeis_id":"","target":"node","first_row":0,"oeis_offset":0,"reversed":false,"trim_zeros":"none","regular_triangle":false,"rows_shown":10,"polytool_commit":"c48629374597092cc41ef4d6273f37991516f984","triangle_example":"Triangle begins:\n  0:    1;\n  1:    3,     1;\n  2:   10,     7,     1;\n  3:   25,    29,    10,     1;\n  4:   75,   120,    65,    14,     1;\n  5:  175,   365,   285,   103,    17,     1;\n  6:  500,  1275,  1275,   635,   165,    21,    1;\n  7: 1125,  3425,  4275,  2820,  1055,   222,   24,   1;\n  8: 3125, 11000, 16250, 13100,  6275,  1820,  310,  28,  1;\n  9: 6875, 27625, 47750, 46450, 27925, 10715, 2610, 386, 31, 1;\n","recurrence_sentence":"The row polynomials satisfy P(n,x) = -(25+50*x+35*x^2+10*x^3+x^4)*P(n-4,x) + (10+10*x+2*x^2)*P(n-2,x) for n >= 4, with P(0,x) = 1, P(1,x) = 3 + x, P(2,x) = 10 + 7*x + x^2, P(3,x) = 25 + 29*x + 10*x^2 + x^3.","mathematica":"P[0, x_] := 1;\nP[1, x_] := 3 + x;\nP[2, x_] := 10 + 7*x + x^2;\nP[3, x_] := 25 + 29*x + 10*x^2 + x^3;\nP[n_, x_] := P[n, x] = Expand[-(25 + 50*x + 35*x^2 + 10*x^3 + x^4)*P[n - 4, x] + (10 + 10*x + 2*x^2)*P[n - 2, x]];\nTable[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten","pari":"P(N) = {my(v = vector(N + 1)); v[1] = 1; v[2] = 3 + x; v[3] = 10 + 7*x + x^2; v[4] = 25 + 29*x + 10*x^2 + x^3; for(n = 4, N, my(i = n + 1); v[i] = -(25 + 50*x + 35*x^2 + 10*x^3 + x^4)*v[i - 4] + (10 + 10*x + 2*x^2)*v[i - 2]); v};\nrow(n, p) = {my(c = Vecrev(p)); c};\nT(N) = {my(v = P(N)); concat(vector(#v, i, row(i - 1, v[i])))};\nT(9)","reading":{"first_row":0,"offset":0,"reversed":false,"trim_zeros":"none","regular_triangle":false}}],"derivation":{"root":"n-4e0b19699f4cf0db","steps":["D+1","x \u21a6 x+1"],"edges":["e-b1073907eb08b9b5","e-6a9ddfa19e5c8e04"],"text":"A026374 \u2192 D+1 \u2192 x \u21a6 x+1"},"headline":"A026374 \u2192 D+1 \u2192 x \u21a6 x+1","chain":null}
