{"id":"n-2038bae4c1bb9757","revision":0,"first_index":1,"fingerprint":"f6-cb11c1d5b8bd540c","name":"one_plus_derivative of basis_map_A013609 of Coefficients of a special case of Poisson-Charlier polynomials.","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":[["1"],["4","2"],["15","18","4"],["66","138","68","8"],["353","1144","1100","288","16"],["2248","10354","16480","10104","1664","32"],["16583","101378","234680","249624","113824","13856","64"],["138750","1070098","3309228","5170144","4175504","1537408","153152","128"],["1297073","12158592","47485752","99236480","117970864","77373952","24081280","2060288","256"],["13395148","148428546","704171328","1857110192","2950472256","2846280288","1578830336","427266816","32179200","512"],["151455535","1942029506","10879899756","34813427504","69703810096","89729310816","73098211136","35232416512","8453498112","569401856","1024"],["1861068762","27156033690","175860508756","663705208440","1610148044352","2608315868576","2831011982784","2000912380032","854350290944","184384562688","11239646208","2048"],["24697985265","404631183768","2978911139972","12981389593952","37100422130784","72726572854144","99195035090752","93273565991424","58320513299968","22382679549952","4393981756416","244697767936","4096"],["352084318160","6405534960754","52900148701472","261817190894632","863105929905792","1990974187373248","3282244885563904","3873607017497728","3218159957936128","1806500369886208","630280861032448","113562469083136","5822638817280","8192"],["5366434103623","107429292852178","984446239131232","5461011400311944","20428261629551920","54304138428768448","105243040288233536","149965353269679744","156200995038723840","116375727493213184","59327712282031104","18989312019990528","3163462702202880","150309602664448","16384"]],"degrees":[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14],"row_sums":["1","6","37","280","2901","40882","730009","15554412","381664537","10558134622","324573461853","10968945597280","404027098348429","16105733275979082","690706566121474993"],"nonn":true,"definitions":[{"position":0,"source":"derived:one_plus_derivative","source_kind":"derived","equality_edge_id":null,"recurrence":{"first_index":1,"initial_polynomials":[["1"],["4","2"],["15","18","4"]],"recurrence":{"denominator":{"coeffs":[["-8"],["4"]]},"terms":[{"coeff":{"coeffs":[["156","1068","2520","2256","480"],["-139","-992","-2424","-2240","-496"],["44","308","744","688","160"],["-5","-32","-72","-64","-16"]]},"deriv_order":0,"offset":3,"sign":"none"},{"coeff":{"coeffs":[["24","204","648","912","480"],["-23","-200","-648","-928","-496"],["8","68","216","304","160"],["-1","-8","-24","-32","-16"]]},"deriv_order":1,"offset":3,"sign":"none"},{"coeff":{"coeffs":[["-68","-292","-376","-160"],["65","304","428","192"],["-18","-90","-144","-72"],["1","6","12","8"]]},"deriv_order":0,"offset":2,"sign":"none"},{"coeff":{"coeffs":[["-8","-68","-184","-160"],["13","96","236","192"],["-6","-42","-96","-72"],["1","6","12","8"]]},"deriv_order":1,"offset":2,"sign":"none"},{"coeff":{"coeffs":[["12","28","-40"],["-19","-40","28"],["7","12","-4"]]},"deriv_order":0,"offset":1,"sign":"none"},{"coeff":{"coeffs":[["-8","-36","-40"],["5","24","28"],["-1","-4","-4"]]},"deriv_order":1,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"}}],"properties":[{"id":185669,"position":0,"claim":"real_rooted","from_index":1,"region_key":"(-inf, 0]","provenance":"edges","claim_data":{"claim":"real_rooted","from_index":1,"region":{"upper":{"closed":true,"value":"0"}}},"label":"Real-rooted","description":"Every row P_n, n >= 1, is zero or real-rooted with all roots in (-inf, 0]","region_exact":null,"edges":["e-aa3d7112894945f0"]},{"id":185670,"position":1,"claim":"one_signed_coefficients","from_index":1,"region_key":null,"provenance":"derived_from_edges","claim_data":{"alternating":false,"claim":"one_signed_coefficients","from_index":1},"label":"Coefficients of one sign","description":"Every nonzero row, n >= 1: the coefficients of P_n have one sign","region_exact":null,"edges":["e-aa3d7112894945f0"]},{"id":185671,"position":2,"claim":"hadamard_kernel","from_index":1,"region_key":null,"provenance":"derived_from_edges","claim_data":{"claim":"hadamard_kernel","from_index":1},"label":"Hadamard kernel (Malo)","description":"Rows P_n, n >= 1, are Hadamard kernels (preserve real-rootedness)","region_exact":null,"edges":["e-aa3d7112894945f0"]}],"region_exact":{},"incoming_edges":[{"id":"e-aa3d7112894945f0","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-2038bae4c1bb9757","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":"058f962bf9b78c1ed261138cf929b368b7f671711eb4b3029d92552e74133c6d","verification_polytool":"replayed","verification_lean":"not_attempted","claims":[{"claim":"real_rooted","from_index":1,"region":{"upper":{"closed":true,"value":"0"}}}],"premises":[{"position":0,"node_id":"n-95c4930a6d47b128","role":"sequence","name":"basis_map_A013609 of Coefficients of a special case of Poisson-Charlier polynomials.","uses":{"claim":"real_rooted","from_index":1,"region":{"upper":{"closed":true,"value":"0"}}},"uses_text":"Every row P_n, n >= 1, 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":1,"oeis_offset":1,"reversed":false,"trim_zeros":"none","regular_triangle":false,"rows_shown":10,"polytool_commit":"c48629374597092cc41ef4d6273f37991516f984","triangle_example":"Triangle begins:\n   1:        1;\n   2:        4,         2;\n   3:       15,        18,         4;\n   4:       66,       138,        68,          8;\n   5:      353,      1144,      1100,        288,         16;\n   6:     2248,     10354,     16480,      10104,       1664,         32;\n   7:    16583,    101378,    234680,     249624,     113824,      13856,         64;\n   8:   138750,   1070098,   3309228,    5170144,    4175504,    1537408,     153152,       128;\n   9:  1297073,  12158592,  47485752,   99236480,  117970864,   77373952,   24081280,   2060288,      256;\n  10: 13395148, 148428546, 704171328, 1857110192, 2950472256, 2846280288, 1578830336, 427266816, 32179200, 512;\n","recurrence_sentence":"The row polynomials satisfy (4*n-8)*P(n,x) = -(5*n^3-44*n^2+139*n-156+(32*n^3-308*n^2+992*n-1068)*x+(72*n^3-744*n^2+2424*n-2520)*x^2+(64*n^3-688*n^2+2240*n-2256)*x^3+(16*n^3-160*n^2+496*n-480)*x^4)*P(n-3,x) - (n^3-8*n^2+23*n-24+(8*n^3-68*n^2+200*n-204)*x+(24*n^3-216*n^2+648*n-648)*x^2+(32*n^3-304*n^2+928*n-912)*x^3+(16*n^3-160*n^2+496*n-480)*x^4)*d/dx P(n-3,x) + (n^3-18*n^2+65*n-68+(6*n^3-90*n^2+304*n-292)*x+(12*n^3-144*n^2+428*n-376)*x^2+(8*n^3-72*n^2+192*n-160)*x^3)*P(n-2,x) + (n^3-6*n^2+13*n-8+(6*n^3-42*n^2+96*n-68)*x+(12*n^3-96*n^2+236*n-184)*x^2+(8*n^3-72*n^2+192*n-160)*x^3)*d/dx P(n-2,x) + (7*n^2-19*n+12+(12*n^2-40*n+28)*x-(4*n^2-28*n+40)*x^2)*P(n-1,x) - (n^2-5*n+8+(4*n^2-24*n+36)*x+(4*n^2-28*n+40)*x^2)*d/dx P(n-1,x) for n >= 4, with P(1,x) = 1, P(2,x) = 4 + 2*x, P(3,x) = 15 + 18*x + 4*x^2.","mathematica":"P[1, x_] := 1;\nP[2, x_] := 4 + 2*x;\nP[3, x_] := 15 + 18*x + 4*x^2;\nP[n_, x_] := P[n, x] = Expand[Together[(-(5*n^3 - 44*n^2 + 139*n - 156 + (32*n^3 - 308*n^2 + 992*n - 1068)*x + (72*n^3 - 744*n^2 + 2424*n - 2520)*x^2 + (64*n^3 - 688*n^2 + 2240*n - 2256)*x^3 + (16*n^3 - 160*n^2 + 496*n - 480)*x^4)*P[n - 3, x] - (n^3 - 8*n^2 + 23*n - 24 + (8*n^3 - 68*n^2 + 200*n - 204)*x + (24*n^3 - 216*n^2 + 648*n - 648)*x^2 + (32*n^3 - 304*n^2 + 928*n - 912)*x^3 + (16*n^3 - 160*n^2 + 496*n - 480)*x^4)*D[P[n - 3, x], x] + (n^3 - 18*n^2 + 65*n - 68 + (6*n^3 - 90*n^2 + 304*n - 292)*x + (12*n^3 - 144*n^2 + 428*n - 376)*x^2 + (8*n^3 - 72*n^2 + 192*n - 160)*x^3)*P[n - 2, x] + (n^3 - 6*n^2 + 13*n - 8 + (6*n^3 - 42*n^2 + 96*n - 68)*x + (12*n^3 - 96*n^2 + 236*n - 184)*x^2 + (8*n^3 - 72*n^2 + 192*n - 160)*x^3)*D[P[n - 2, x], x] + (7*n^2 - 19*n + 12 + (12*n^2 - 40*n + 28)*x - (4*n^2 - 28*n + 40)*x^2)*P[n - 1, x] - (n^2 - 5*n + 8 + (4*n^2 - 24*n + 36)*x + (4*n^2 - 28*n + 40)*x^2)*D[P[n - 1, x], x])/(4*n - 8)]];\nTable[CoefficientList[P[n, x], x], {n, 1, 10}] // Flatten","pari":"P(N) = {my(v = vector(N)); v[1] = 1; v[2] = 4 + 2*x; v[3] = 15 + 18*x + 4*x^2; for(n = 4, N, my(i = n); v[i] = (-(5*n^3 - 44*n^2 + 139*n - 156 + (32*n^3 - 308*n^2 + 992*n - 1068)*x + (72*n^3 - 744*n^2 + 2424*n - 2520)*x^2 + (64*n^3 - 688*n^2 + 2240*n - 2256)*x^3 + (16*n^3 - 160*n^2 + 496*n - 480)*x^4)*v[i - 3] - (n^3 - 8*n^2 + 23*n - 24 + (8*n^3 - 68*n^2 + 200*n - 204)*x + (24*n^3 - 216*n^2 + 648*n - 648)*x^2 + (32*n^3 - 304*n^2 + 928*n - 912)*x^3 + (16*n^3 - 160*n^2 + 496*n - 480)*x^4)*deriv(v[i - 3], x) + (n^3 - 18*n^2 + 65*n - 68 + (6*n^3 - 90*n^2 + 304*n - 292)*x + (12*n^3 - 144*n^2 + 428*n - 376)*x^2 + (8*n^3 - 72*n^2 + 192*n - 160)*x^3)*v[i - 2] + (n^3 - 6*n^2 + 13*n - 8 + (6*n^3 - 42*n^2 + 96*n - 68)*x + (12*n^3 - 96*n^2 + 236*n - 184)*x^2 + (8*n^3 - 72*n^2 + 192*n - 160)*x^3)*deriv(v[i - 2], x) + (7*n^2 - 19*n + 12 + (12*n^2 - 40*n + 28)*x - (4*n^2 - 28*n + 40)*x^2)*v[i - 1] - (n^2 - 5*n + 8 + (4*n^2 - 24*n + 36)*x + (4*n^2 - 28*n + 40)*x^2)*deriv(v[i - 1], x))/(4*n - 8)); v};\nrow(n, p) = {my(c = Vecrev(p)); c};\nT(N) = {my(v = P(N)); concat(vector(#v, i, row(i, v[i])))};\nT(10)","reading":{"first_row":1,"offset":1,"reversed":false,"trim_zeros":"none","regular_triangle":false}}],"derivation":{"root":"n-c9dce8c25621fdea","steps":["basis A013609","D+1"],"edges":["e-0705555781f6cbb3","e-aa3d7112894945f0"],"text":"A046716 \u2192 basis A013609 \u2192 D+1"},"headline":"A046716 \u2192 basis A013609 \u2192 D+1","chain":null}