{"id":"n-cb146e943380d277","revision":0,"first_index":0,"fingerprint":"f6-95f8f0bdb419f7a5","name":"derivative of A triangle of numbers related to triangle A030528; array a(n,m), read by rows (1 <= m <= n).","polytool_family":null,"interpretation":null,"rows_observed":15,"degree_pattern":"affine","degree_slope":1,"degree_intercept":1,"palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":false,"rows":[["3","2"],["3","12","3"],["0","30","30","4"],["0","30","135","60","5"],["0","0","315","420","105","6"],["0","0","315","1680","1050","168","7"],["0","0","0","3780","6300","2268","252","8"],["0","0","0","3780","23625","18900","4410","360","9"],["0","0","0","0","51975","103950","48510","7920","495","10"],["0","0","0","0","51975","374220","363825","110880","13365","660","11"],["0","0","0","0","0","810810","1891890","1081080","231660","21450","858","12"],["0","0","0","0","0","810810","6621615","7567560","2837835","450450","33033","1092","13"],["0","0","0","0","0","0","14189175","37837800","25540515","6756750","825825","49140","1365","14"],["0","0","0","0","0","0","14189175","129729600","170270100","75675600","14864850","1441440","70980","1680","15"],["0","0","0","0","0","0","0","275675400","827026200","643242600","202161960","30630600","2413320","99960","2040","16"]],"degrees":[1,2,3,4,5,6,7,8,9,10,11,12,13,14,15],"row_sums":["5","18","64","230","846","3220","12608","51084","212860","914936","4037760","18322408","85200584","406243440","1981252096"],"nonn":true,"definitions":[{"position":0,"source":"derived:derivative","source_kind":"derived","equality_edge_id":null,"recurrence":{"first_index":0,"initial_polynomials":[["3","2"]],"recurrence":{"denominator":{"coeffs":[["0","-2"],["1","-2"],["1","0"]]},"terms":[{"coeff":{"coeffs":[["0","-6","-4"],["2","3","-2"],["1","3","0"],["-1","0","0"]]},"deriv_order":0,"offset":1,"sign":"none"},{"coeff":{"coeffs":[["0","0","-8"],["0","2","-4"],["0","2","0"]]},"deriv_order":1,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"}}],"properties":[{"id":176260,"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-281dca6f54d4bea4"]},{"id":176261,"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-281dca6f54d4bea4"]},{"id":176262,"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-281dca6f54d4bea4"]}],"region_exact":{},"incoming_edges":[{"id":"e-281dca6f54d4bea4","operation":{"index_shift":0,"operation":"preserver_image","operator":"{\"shape\":\"derivative_polynomial\",\"coefficients\":[{\"numerator\":[\"0\"],\"denominator\":[\"1\"]},{\"numerator\":[\"1\"],\"denominator\":[\"1\"]}]}"},"operation_detail":"derivative_polynomial, shift 0","conclusion_id":"n-cb146e943380d277","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":"3b9331f820a1e0f54ab485d2d0f99efc7ef3d8525064450d40fb6ff77555020c","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-47e3a1a59edf16da","role":"sequence","name":"A triangle of numbers related to triangle A030528; array a(n,m), read by rows (1 <= m <= n).","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: 3,  2;\n  1: 3, 12,   3;\n  2: 0, 30,  30,    4;\n  3: 0, 30, 135,   60,     5;\n  4: 0,  0, 315,  420,   105,      6;\n  5: 0,  0, 315, 1680,  1050,    168,      7;\n  6: 0,  0,   0, 3780,  6300,   2268,    252,      8;\n  7: 0,  0,   0, 3780, 23625,  18900,   4410,    360,     9;\n  8: 0,  0,   0,    0, 51975, 103950,  48510,   7920,   495,  10;\n  9: 0,  0,   0,    0, 51975, 374220, 363825, 110880, 13365, 660, 11;\n","recurrence_sentence":"The row polynomials satisfy (n^2+n-(2*n+2)*x)*P(n,x) = -(n^3-n^2-2*n-(3*n^2+3*n-6)*x+(2*n+4)*x^2)*P(n-1,x) + ((2*n^2+2*n)*x-(4*n+8)*x^2)*d/dx P(n-1,x) for n >= 1, with P(0,x) = 3 + 2*x.","mathematica":"P[0, x_] := 3 + 2*x;\nP[n_, x_] := P[n, x] = Expand[Together[(-(n^3 - n^2 - 2*n - (3*n^2 + 3*n - 6)*x + (2*n + 4)*x^2)*P[n - 1, x] + ((2*n^2 + 2*n)*x - (4*n + 8)*x^2)*D[P[n - 1, x], x])/(n^2 + n - (2*n + 2)*x)]];\nTable[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten","pari":"P(N) = {my(v = vector(N + 1)); v[1] = 3 + 2*x; for(n = 1, N, my(i = n + 1); v[i] = (-(n^3 - n^2 - 2*n - (3*n^2 + 3*n - 6)*x + (2*n + 4)*x^2)*v[i - 1] + ((2*n^2 + 2*n)*x - (4*n + 8)*x^2)*deriv(v[i - 1], x))/(n^2 + n - (2*n + 2)*x)); 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-47e3a1a59edf16da","steps":["D"],"edges":["e-281dca6f54d4bea4"],"text":"A049403 \u2192 D"},"headline":"A049403 \u2192 D","chain":null}