{"id":"n-4073075195c11dd3","revision":0,"first_index":1,"fingerprint":"f6-353fe824af633f29","name":"one_plus_derivative of basis_map_A013609 of Triangle read by rows: T(n, k) = binomial(n+k-1, 2*k-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"],["5","2"],["20","20","4"],["71","122","60","8"],["235","588","492","160","16"],["744","2472","3000","1648","400","32"],["2285","9516","15324","12160","4960","960","64"],["6865","34462","69612","73160","42960","13920","2240","128"],["20284","119356","291020","383424","297600","138432","37184","5120","256"],["59155","399574","1143816","1819280","1768368","1085280","417536","95744","11520","512"],["170711","1302456","4286760","8008864","9394000","7211904","3654784","1198080","239616","25600","1024"],["488400","4155216","15470832","33255520","45817120","42418880","26861184","11582208","3306240","586240","56320","2048"],["1387225","13024056","54149352","131775520","208927120","227134080","173330304","93293568","34986240","8842240","1408000","122880","4096"],["3916061","40223066","184783560","502549904","902358000","1128839904","1011108864","655380480","306510336","101657600","23046144","3330048","266240","8192"],["10996580","122673764","617272276","1856422080","3726776448","5280338880","5441249856","4137729024","2328817920","962429952","286049280","58785792","7774208","573440","16384"]],"degrees":[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14],"row_sums":["1","7","44","261","1491","8296","45269","243347","1292676","6800785","35493799","184000208","948384681","4863978399","24837905884"],"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":[["-52","0","16"],["84","0","-16"],["-48","0","0"],["16","0","0"]]},"terms":[{"coeff":{"coeffs":[["0","0","0","0"],["34","-12","-40","-16"],["0","0","0","0"],["24","16","0","0"]]},"deriv_order":0,"offset":1,"sign":"none"},{"coeff":{"coeffs":[["5","-28","-112","-80","-16"],["-20","-48","-16","0","0"],["20","48","16","0","0"]]},"deriv_order":1,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"}}],"properties":[{"id":178138,"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-dc3766b87ff15105"]},{"id":178139,"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-dc3766b87ff15105"]},{"id":178140,"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-dc3766b87ff15105"]},{"id":178141,"position":3,"claim":"interlacing","from_index":2,"region_key":null,"provenance":"edges","claim_data":{"claim":"interlacing","from_index":2,"kind":"weak","orientation":"unordered"},"label":"Interlacing of consecutive rows","description":"Rows P_(m-1), P_m interlace (weak, unordered) for every m >= 2","region_exact":null,"edges":["e-dc3766b87ff15105"]}],"region_exact":{},"incoming_edges":[{"id":"e-dc3766b87ff15105","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-4073075195c11dd3","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":"ebcf0ec06583946b79d82bfafc727705c8f9fb7444b04d92de57d349d2a2cc1b","verification_polytool":"replayed","verification_lean":"not_attempted","claims":[{"claim":"real_rooted","from_index":1,"region":{"upper":{"closed":true,"value":"0"}}},{"claim":"interlacing","from_index":2,"kind":"weak","orientation":"unordered"}],"premises":[{"position":0,"node_id":"n-8fd7b2e367875124","role":"sequence","name":"basis_map_A013609 of Triangle read by rows: T(n, k) = binomial(n+k-1, 2*k-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":[],"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:     5,      2;\n   3:    20,     20,       4;\n   4:    71,    122,      60,       8;\n   5:   235,    588,     492,     160,      16;\n   6:   744,   2472,    3000,    1648,     400,      32;\n   7:  2285,   9516,   15324,   12160,    4960,     960,     64;\n   8:  6865,  34462,   69612,   73160,   42960,   13920,   2240,   128;\n   9: 20284, 119356,  291020,  383424,  297600,  138432,  37184,  5120,   256;\n  10: 59155, 399574, 1143816, 1819280, 1768368, 1085280, 417536, 95744, 11520, 512;\n","recurrence_sentence":"The row polynomials satisfy (16*n^3-48*n^2+84*n-52-(16*n-16)*x^2)*P(n,x) = (24*n^3+34*n+(16*n^3-12*n)*x-40*n*x^2-16*n*x^3)*P(n-1,x) + (20*n^2-20*n+5+(48*n^2-48*n-28)*x+(16*n^2-16*n-112)*x^2-80*x^3-16*x^4)*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[((24*n^3 + 34*n + (16*n^3 - 12*n)*x - 40*n*x^2 - 16*n*x^3)*P[n - 1, x] + (20*n^2 - 20*n + 5 + (48*n^2 - 48*n - 28)*x + (16*n^2 - 16*n - 112)*x^2 - 80*x^3 - 16*x^4)*D[P[n - 1, x], x])/(16*n^3 - 48*n^2 + 84*n - 52 - (16*n - 16)*x^2)]];\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] = ((24*n^3 + 34*n + (16*n^3 - 12*n)*x - 40*n*x^2 - 16*n*x^3)*v[i - 1] + (20*n^2 - 20*n + 5 + (48*n^2 - 48*n - 28)*x + (16*n^2 - 16*n - 112)*x^2 - 80*x^3 - 16*x^4)*deriv(v[i - 1], x))/(16*n^3 - 48*n^2 + 84*n - 52 - (16*n - 16)*x^2)); 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-976cdb6e4a693ec8","steps":["basis A013609","D+1"],"edges":["e-567d6f79aa6a87e5","e-dc3766b87ff15105"],"text":"A078812 \u2192 basis A013609 \u2192 D+1"},"headline":"A078812 \u2192 basis A013609 \u2192 D+1","chain":null}
