{"id":"n-09ca1d900013a9af","revision":0,"first_index":1,"fingerprint":"f6-10b1f054401300a3","name":"one_plus_derivative of basis_map_A013611 of Triangle read by rows, T(n,k) = T(n-1,k) + T(n-2,k) + T(n-1,k-1) + T(n-2,k-1) and T(0,0) = 1.","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"],["6","4"],["26","52","16"],["96","336","320","64"],["332","1664","2800","1728","256"],["1096","7184","17504","19136","8704","1024"],["3512","28464","91168","144960","116736","41984","4096"],["11008","106368","422912","886784","1040384","661504","196608","16384"],["33936","380928","1810560","4724224","7321856","6785024","3559424","901120","65536"],["103264","1320768","7307520","22836736","44109312","54043648","41312256","18432000","4063232","262144"],["310944","4464448","28190208","102702592","237890048","363570176","367579136","238862336","92667904","18087936","1048576"],["928256","14784256","104928256","436802560","1180762112","2165903360","2732867584","2348859392","1326448640","455081984","79691776","4194304"],["2751168","48138240","379387136","1776923648","5493470208","11768848384","17832554496","19132661760","14288551936","7131365376","2192572416","348127232","16777216"],["8104064","154535168","1339179520","6970848256","24265562112","59493380096","105338527744","135766589440","126579113984","83525894144","37335597056","10397679616","1509949440","67108864"],["23746432","490152704","4632481280","26533014528","102716123136","283726417920","575204925440","866607185920","970766417920","799678136320","472458985472","191180570624","48653926400","6509559808","268435456"]],"degrees":[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14],"row_sums":["1","10","94","816","6780","54648","430920","3341952","25582608","193790880","1455374304","10851252480","80412129216","592752069504","4349450079360"],"nonn":true,"definitions":[{"position":0,"source":"derived:one_plus_derivative","source_kind":"derived","equality_edge_id":null,"recurrence":{"first_index":1,"initial_polynomials":[["1"]],"recurrence":{"denominator":{"coeffs":[["3","22","12","-8"],["1","-30","-20","8"],["-6","8","8","0"],["2","0","0","0"]]},"terms":[{"coeff":{"coeffs":[["18","-72","-80","32","32"],["-74","-4","72","16","0"],["36","8","-16","0","0"],["-4","0","0","0","0"]]},"deriv_order":0,"offset":1,"sign":"none"},{"coeff":{"coeffs":[["3","-10","-32","32","80","32"],["-12","-8","72","96","32","0"],["6","16","8","0","0","0"]]},"deriv_order":1,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"}}],"properties":[{"id":182119,"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-68c1b1a37ae79705"]},{"id":182120,"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-68c1b1a37ae79705"]},{"id":182121,"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-68c1b1a37ae79705"]}],"region_exact":{},"incoming_edges":[{"id":"e-68c1b1a37ae79705","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-09ca1d900013a9af","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":"0643b3455641e7e4b9c576b12ccadfce8bc1e603a4e1275966b235c638bd8074","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-2156f740e0c97180","role":"sequence","name":"basis_map_A013611 of Triangle read by rows, T(n,k) = T(n-1,k) + T(n-2,k) + T(n-1,k-1) + T(n-2,k-1) and T(0,0) = 1.","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":[{"id":"l-e4e810119e327a1b","kind":"fingerprint_related","node_a_id":"n-09ca1d900013a9af","node_b_id":"n-dfe62937de87f91f","fingerprint":"f6-10b1f054401300a3","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-10b1f054401300a3"},"other_id":"n-dfe62937de87f91f","other":{"id":"n-dfe62937de87f91f","name":"one_plus_derivative of basis_map_A013609 of Riordan array (1, 2+2x).","first_index":0,"fingerprint":"f6-10b1f054401300a3","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":0,"preview_rows":[["1"],["6","4"],["26","52","16"]],"card_rows":[["1"],["6","4"],["26","52","16"],["96","336","320","64"]],"row_sums_preview":["1","10","94","816","6780","54648","430920","3341952"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-f5a001953ee28e8f","steps":["basis A013609","D+1"],"edges":["e-1a65135df7496975","e-05fbf3619c3db6b2"],"text":"A099040 \u2192 basis A013609 \u2192 D+1"},"headline":"A099040 \u2192 basis A013609 \u2192 D+1","chain":null}}],"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:      6,       4;\n   3:     26,      52,      16;\n   4:     96,     336,     320,       64;\n   5:    332,    1664,    2800,     1728,      256;\n   6:   1096,    7184,   17504,    19136,     8704,     1024;\n   7:   3512,   28464,   91168,   144960,   116736,    41984,     4096;\n   8:  11008,  106368,  422912,   886784,  1040384,   661504,   196608,    16384;\n   9:  33936,  380928, 1810560,  4724224,  7321856,  6785024,  3559424,   901120,   65536;\n  10: 103264, 1320768, 7307520, 22836736, 44109312, 54043648, 41312256, 18432000, 4063232, 262144;\n","recurrence_sentence":"The row polynomials satisfy (2*n^3-6*n^2+n+3+(8*n^2-30*n+22)*x+(8*n^2-20*n+12)*x^2+(8*n-8)*x^3)*P(n,x) = -(4*n^3-36*n^2+74*n-18-(8*n^2-4*n-72)*x+(16*n^2-72*n+80)*x^2-(16*n+32)*x^3-32*x^4)*P(n-1,x) + (6*n^2-12*n+3+(16*n^2-8*n-10)*x+(8*n^2+72*n-32)*x^2+(96*n+32)*x^3+(32*n+80)*x^4+32*x^5)*d/dx P(n-1,x) for n >= 2, with P(1,x) = 1.","mathematica":"P[1, x_] := 1;\nP[n_, x_] := P[n, x] = Expand[Together[(-(4*n^3 - 36*n^2 + 74*n - 18 - (8*n^2 - 4*n - 72)*x + (16*n^2 - 72*n + 80)*x^2 - (16*n + 32)*x^3 - 32*x^4)*P[n - 1, x] + (6*n^2 - 12*n + 3 + (16*n^2 - 8*n - 10)*x + (8*n^2 + 72*n - 32)*x^2 + (96*n + 32)*x^3 + (32*n + 80)*x^4 + 32*x^5)*D[P[n - 1, x], x])/(2*n^3 - 6*n^2 + n + 3 + (8*n^2 - 30*n + 22)*x + (8*n^2 - 20*n + 12)*x^2 + (8*n - 8)*x^3)]];\nTable[CoefficientList[P[n, x], x], {n, 1, 10}] // Flatten","pari":"P(N) = {my(v = vector(N)); v[1] = 1; for(n = 2, N, my(i = n); v[i] = (-(4*n^3 - 36*n^2 + 74*n - 18 - (8*n^2 - 4*n - 72)*x + (16*n^2 - 72*n + 80)*x^2 - (16*n + 32)*x^3 - 32*x^4)*v[i - 1] + (6*n^2 - 12*n + 3 + (16*n^2 - 8*n - 10)*x + (8*n^2 + 72*n - 32)*x^2 + (96*n + 32)*x^3 + (32*n + 80)*x^4 + 32*x^5)*deriv(v[i - 1], x))/(2*n^3 - 6*n^2 + n + 3 + (8*n^2 - 30*n + 22)*x + (8*n^2 - 20*n + 12)*x^2 + (8*n - 8)*x^3)); 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-c72653d9ab0ee8be","steps":["basis A013611","D+1"],"edges":["e-13a04d0165daacbb","e-68c1b1a37ae79705"],"text":"A063967 \u2192 basis A013611 \u2192 D+1"},"headline":"A063967 \u2192 basis A013611 \u2192 D+1","chain":null}