{"id":"n-61fe0d9f1ff3ff7f","revision":0,"first_index":0,"fingerprint":"f6-9342e2e07a2452d6","name":"one_plus_derivative of Triangle T(n,k) read by rows: associated Stirling numbers of first kind (n >= 2, 1 <= k <= floor(n/2)).","polytool_family":null,"interpretation":null,"rows_observed":15,"degree_pattern":"half_affine","degree_slope":1,"degree_intercept":0,"palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":false,"rows":[["1"],["2"],["9","3"],["44","20"],["250","160","15"],["1644","1344","210"],["12348","12068","2695","105"],["104544","117088","33992","2520"],["986256","1230696","435960","47880","945"],["10265760","13994640","5794800","843920","34650"],["116915040","171646992","80620716","14563780","918225","10395"],["1446526080","2262596544","1179530352","252766800","21561540","540540"],["19323757440","31934627712","18180522360","4478321848","480890410","19099080","135135"],["277238626560","480881495808","295269169104","81692570960","10535224700","576816240","9459450"],["4251984710400","7699529848320","5049341135280","1541904283920","231044934120","16139923800","430404975","2027025"]],"degrees":[0,0,1,1,2,2,3,3,4,4,5,5,6,6,7],"row_sums":["1","2","12","64","425","3198","27216","258144","2701737","30933770","384675148","5163521856","74417353985","1146203362822","18790377267840"],"nonn":true,"definitions":[{"position":0,"source":"derived:one_plus_derivative","source_kind":"derived","equality_edge_id":null,"recurrence":{"first_index":0,"initial_polynomials":[["1"],["2"],["9","3"],["44","20"]],"recurrence":{"denominator":{"coeffs":[["0"],["1"]]},"terms":[{"coeff":{"coeffs":[["0","0","0"],["0","0","1"],["0","0","0"],["0","0","-1"]]},"deriv_order":0,"offset":4,"sign":"none"},{"coeff":{"coeffs":[["0","0"],["0","2"],["0","0"],["0","-2"]]},"deriv_order":0,"offset":3,"sign":"none"},{"coeff":{"coeffs":[["0","0"],["1","2"],["0","2"],["-1","0"]]},"deriv_order":0,"offset":2,"sign":"none"},{"coeff":{"coeffs":[["-1"],["1"],["2"]]},"deriv_order":0,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"}}],"properties":[{"id":181460,"position":0,"claim":"real_rooted","from_index":0,"region_key":"(-inf, 0]","provenance":"edges","claim_data":{"claim":"real_rooted","from_index":0,"region":{"upper":{"closed":true,"value":"0"}}},"label":"Real-rooted","description":"Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, 0]","region_exact":null,"edges":["e-f87f7217fdb430c9"]},{"id":181461,"position":1,"claim":"one_signed_coefficients","from_index":0,"region_key":null,"provenance":"derived_from_edges","claim_data":{"alternating":false,"claim":"one_signed_coefficients","from_index":0},"label":"Coefficients of one sign","description":"Every nonzero row, n >= 0: the coefficients of P_n have one sign","region_exact":null,"edges":["e-f87f7217fdb430c9"]},{"id":181462,"position":2,"claim":"hadamard_kernel","from_index":0,"region_key":null,"provenance":"derived_from_edges","claim_data":{"claim":"hadamard_kernel","from_index":0},"label":"Hadamard kernel (Malo)","description":"Rows P_n, n >= 0, are Hadamard kernels (preserve real-rootedness)","region_exact":null,"edges":["e-f87f7217fdb430c9"]}],"region_exact":{},"incoming_edges":[{"id":"e-f87f7217fdb430c9","operation":{"index_shift":0,"operation":"preserver_image","operator":"{\"shape\":\"derivative_polynomial\",\"coefficients\":[{\"numerator\":[\"1\"],\"denominator\":[\"1\"]},{\"numerator\":[\"1\"],\"denominator\":[\"1\"]}]}"},"operation_detail":"derivative_polynomial, shift 0","conclusion_id":"n-61fe0d9f1ff3ff7f","premise_count":1,"theorem":"Laguerre's theorem on differential operators: if phi is a real polynomial with only real roots, then phi(D) maps every real-rooted polynomial to a real-rooted polynomial or zero (write phi(D) = C D^j prod (1 + c_i D); D preserves real-rootedness by Rolle's theorem and p + c p' is real-rooted for every real c by Rolle's theorem applied to e^(x/c) p; Polya--Schur 1914); when the coefficients of phi have one sign, coefficients of one sign stay of one sign; applied to phi(n, .) for each n","lean_route":null,"lean_declarations":["RealRooted.splits_add_C_mul_derivative_all","RealRooted.splits_C_mul_add_derivative","Polynomial.Splits.derivative"],"certificate_hash":"d8b98b13121cd6ec8636254c940830a4c201805bd9d78dc1911c7ca523d32e5a","verification_polytool":"replayed","verification_lean":"not_attempted","claims":[{"claim":"real_rooted","from_index":0,"region":{"upper":{"closed":true,"value":"0"}}}],"premises":[{"position":0,"node_id":"n-f13c6b611f4ce3bb","role":"sequence","name":"Triangle T(n,k) read by rows: associated Stirling numbers of first kind (n >= 2, 1 <= k <= floor(n/2)).","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:        2;\n  2:        9,        3;\n  3:       44,       20;\n  4:      250,      160,      15;\n  5:     1644,     1344,     210;\n  6:    12348,    12068,    2695,    105;\n  7:   104544,   117088,   33992,   2520;\n  8:   986256,  1230696,  435960,  47880,   945;\n  9: 10265760, 13994640, 5794800, 843920, 34650;\n","recurrence_sentence":"The row polynomials satisfy n*P(n,x) = -(n^3-n)*x^2*P(n-4,x) - (2*n^3-2*n)*x*P(n-3,x) - (n^3-n-(2*n^2+2*n)*x)*P(n-2,x) + (2*n^2+n-1)*P(n-1,x) for n >= 4, with P(0,x) = 1, P(1,x) = 2, P(2,x) = 9 + 3*x, P(3,x) = 44 + 20*x.","mathematica":"P[0, x_] := 1;\nP[1, x_] := 2;\nP[2, x_] := 9 + 3*x;\nP[3, x_] := 44 + 20*x;\nP[n_, x_] := P[n, x] = Expand[Together[(-(n^3 - n)*x^2*P[n - 4, x] - (2*n^3 - 2*n)*x*P[n - 3, x] - (n^3 - n - (2*n^2 + 2*n)*x)*P[n - 2, x] + (2*n^2 + n - 1)*P[n - 1, x])/(n)]];\nTable[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten","pari":"P(N) = {my(v = vector(N + 1)); v[1] = 1; v[2] = 2; v[3] = 9 + 3*x; v[4] = 44 + 20*x; for(n = 4, N, my(i = n + 1); v[i] = (-(n^3 - n)*x^2*v[i - 4] - (2*n^3 - 2*n)*x*v[i - 3] - (n^3 - n - (2*n^2 + 2*n)*x)*v[i - 2] + (2*n^2 + n - 1)*v[i - 1])/(n)); 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-f13c6b611f4ce3bb","steps":["D+1"],"edges":["e-f87f7217fdb430c9"],"text":"A008306 \u2192 D+1"},"headline":"A008306 \u2192 D+1","chain":null}