{"id":"n-38cb1bd2c71c1a9f","revision":0,"first_index":0,"fingerprint":"f6-35cdc7685b139f0b","name":"one_plus_derivative of one_plus_derivative of Stirling2 triangle with scaled diagonals (powers of 2).","polytool_family":null,"interpretation":null,"rows_observed":15,"degree_pattern":"affine","degree_slope":1,"degree_intercept":0,"palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":false,"rows":[["1"],["4","1"],["18","10","1"],["88","82","18","1"],["456","640","232","28","1"],["2464","4936","2640","520","40","1"],["13728","38080","28336","8120","1010","54","1"],["78208","295072","295008","115696","20720","1778","70","1"],["452736","2298880","3020032","1561728","378336","46368","2912","88","1"],["2650624","18004096","30620160","20373760","6424320","1055712","94080","4512","108","1"],["15641088","141644800","308720896","259853440","103940320","22020768","2613072","176880","6690","130","1"],["92796928","1118566912","3102325248","3263324416","1626000640","432499936","65677920","5886672","312840","9570","154","1"],["552585216","8859934720","31113951232","40536624128","24827069696","8135652096","1538927104","175471296","12289992","526240","13288","180","1"],["3298729984","70343673856","311683706880","499582720000","372339968000","148221801728","34253739520","4834745344","428684256","24096072","848848","17992","208","1"],["19725262848","559517777920","3120102240256","6121246709760","5508908321280","2635639598592","733423378688","125642682880","13720358080","972331360","44812768","1321320","23842","238","1"]],"degrees":[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14],"row_sums":["1","5","29","189","1357","10601","89329","806553","7761081","79227373","854618085","9707401237","115753045189","1445012732689","18838944819833"],"nonn":true,"definitions":[{"position":0,"source":"derived:one_plus_derivative","source_kind":"derived","equality_edge_id":null,"recurrence":{"first_index":0,"initial_polynomials":[["1"],["4","1"],["18","10","1"]],"recurrence":{"terms":[{"coeff":{"coeffs":[["48","-28","4","3"]]},"deriv_order":0,"offset":3,"sign":"none"},{"coeff":{"coeffs":[["0","-16","-4","8"]]},"deriv_order":1,"offset":3,"sign":"none"},{"coeff":{"coeffs":[["0","0","-8","4"]]},"deriv_order":2,"offset":3,"sign":"none"},{"coeff":{"coeffs":[["-44","20","1"]]},"deriv_order":0,"offset":2,"sign":"none"},{"coeff":{"coeffs":[["0","8","8"]]},"deriv_order":1,"offset":2,"sign":"none"},{"coeff":{"coeffs":[["0","0","4"]]},"deriv_order":2,"offset":2,"sign":"none"},{"coeff":{"coeffs":[["12","-3"]]},"deriv_order":0,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"}}],"properties":[{"id":181441,"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-c142f8422993e1e0"]},{"id":181442,"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-c142f8422993e1e0"]},{"id":181443,"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-c142f8422993e1e0"]}],"region_exact":{},"incoming_edges":[{"id":"e-c142f8422993e1e0","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-38cb1bd2c71c1a9f","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":"c8b31987e170f3eb4513a7d6bcb91d7c41eaafe6afcf99a6a47f9b17bd1b1436","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-df88941ef8650983","role":"sequence","name":"one_plus_derivative of Stirling2 triangle with scaled diagonals (powers of 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:       4,        1;\n  2:      18,       10,        1;\n  3:      88,       82,       18,        1;\n  4:     456,      640,      232,       28,       1;\n  5:    2464,     4936,     2640,      520,      40,       1;\n  6:   13728,    38080,    28336,     8120,    1010,      54,     1;\n  7:   78208,   295072,   295008,   115696,   20720,    1778,    70,    1;\n  8:  452736,  2298880,  3020032,  1561728,  378336,   46368,  2912,   88,   1;\n  9: 2650624, 18004096, 30620160, 20373760, 6424320, 1055712, 94080, 4512, 108, 1;\n","recurrence_sentence":"The row polynomials satisfy P(n,x) = (48-28*x+4*x^2+3*x^3)*P(n-3,x) - (16*x+4*x^2-8*x^3)*d/dx P(n-3,x) - (8*x^2-4*x^3)*d^2/dx^2 P(n-3,x) - (44-20*x-x^2)*P(n-2,x) + (8*x+8*x^2)*d/dx P(n-2,x) + 4*x^2*d^2/dx^2 P(n-2,x) + (12-3*x)*P(n-1,x) for n >= 3, with P(0,x) = 1, P(1,x) = 4 + x, P(2,x) = 18 + 10*x + x^2.","mathematica":"P[0, x_] := 1;\nP[1, x_] := 4 + x;\nP[2, x_] := 18 + 10*x + x^2;\nP[n_, x_] := P[n, x] = Expand[(48 - 28*x + 4*x^2 + 3*x^3)*P[n - 3, x] - (16*x + 4*x^2 - 8*x^3)*D[P[n - 3, x], x] - (8*x^2 - 4*x^3)*D[P[n - 3, x], {x, 2}] - (44 - 20*x - x^2)*P[n - 2, x] + (8*x + 8*x^2)*D[P[n - 2, x], x] + 4*x^2*D[P[n - 2, x], {x, 2}] + (12 - 3*x)*P[n - 1, x]];\nTable[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten","pari":"P(N) = {my(v = vector(N + 1)); v[1] = 1; v[2] = 4 + x; v[3] = 18 + 10*x + x^2; for(n = 3, N, my(i = n + 1); v[i] = (48 - 28*x + 4*x^2 + 3*x^3)*v[i - 3] - (16*x + 4*x^2 - 8*x^3)*deriv(v[i - 3], x) - (8*x^2 - 4*x^3)*deriv(deriv(v[i - 3], x), x) - (44 - 20*x - x^2)*v[i - 2] + (8*x + 8*x^2)*deriv(v[i - 2], x) + 4*x^2*deriv(deriv(v[i - 2], x), x) + (12 - 3*x)*v[i - 1]); 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-a7e9ebb9860a9dbe","steps":["D+1","D+1"],"edges":["e-9fe96d700c9448ac","e-c142f8422993e1e0"],"text":"A075497 \u2192 D+1 \u2192 D+1"},"headline":"A075497 \u2192 D+1 \u2192 D+1","chain":null}