{"id":"n-5d685a1643c9efb9","revision":0,"first_index":0,"fingerprint":"f6-13be81c621b680ef","name":"one_plus_derivative of basis_map_A008277 of Riordan array (1-x,x/(1-x)).","polytool_family":null,"interpretation":null,"rows_observed":15,"degree_pattern":"affine","degree_slope":1,"degree_intercept":2,"palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":false,"rows":[["4","5","1"],["12","24","10","1"],["36","108","74","16","1"],["108","468","478","170","23","1"],["324","1980","2858","1498","331","31","1"],["972","8244","16270","11846","3815","579","40","1"],["2916","33948","89594","87346","38551","8447","939","50","1"],["8748","138708","481918","613670","357203","106127","16898","1439","61","1"],["26244","563580","2548298","4163938","3114091","1206219","258209","31288","2110","73","1"],["78732","2280564","13305070","27531926","25962575","12763843","3530100","571089","54498","2986","86","1"],["236196","9200988","68805914","178496626","209269951","128073319","44534743","9240990","1170567","90330","4104","100","1"],["708588","37040148","353230558","1139785670","1643386283","1233856503","528886006","136944643","22117227","2254527","143682","5504","115","1"],["2125764","148869180","1803192938","7191944578","12643489651","11514238307","5993830557","1898332436","380234140","49171551","4122393","220738","7229","131","1"],["6377292","597602484","9164833870","44954860406","95696372135","104757396107","65458713320","24977154917","6080907976","970292752","102762660","7212725","329173","9325","148","1"],["19131876","2396787228","46421771834","278893996306","714829465351","933755540991","693885815987","315230262490","91867142653","17724421000","2306207332","203740810","12150320","478373","11841","166","1"]],"degrees":[2,3,4,5,6,7,8,9,10,11,12,13,14,15,16],"row_sums":["10","47","235","1248","7023","41767","261792","1724773","11914051","86081470","649123829","5098359455","41629779594","352774825291","3097546924559"],"nonn":true,"definitions":[{"position":0,"source":"derived:one_plus_derivative","source_kind":"derived","equality_edge_id":null,"recurrence":{"first_index":0,"initial_polynomials":[["4","5","1"]],"recurrence":{"terms":[{"coeff":{"coeffs":[["3","1"]]},"deriv_order":0,"offset":1,"sign":"none"},{"coeff":{"coeffs":[["0","1"]]},"deriv_order":1,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"}}],"properties":[{"id":180888,"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-479446aed5ce9d3c","e-e1c5d532853b1a59"]},{"id":180889,"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-479446aed5ce9d3c","e-e1c5d532853b1a59"]},{"id":180890,"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-479446aed5ce9d3c","e-e1c5d532853b1a59"]},{"id":180891,"position":3,"claim":"interlacing","from_index":1,"region_key":null,"provenance":"edges","claim_data":{"claim":"interlacing","from_index":1,"kind":"weak","orientation":"unordered"},"label":"Interlacing of consecutive rows","description":"Rows P_(m-1), P_m interlace (weak, unordered) for every m >= 1","region_exact":null,"edges":["e-479446aed5ce9d3c"]}],"region_exact":{},"incoming_edges":[{"id":"e-e1c5d532853b1a59","operation":{"map":"A008277","operation":"basis_map"},"operation_detail":"A008277","conclusion_id":"n-5d685a1643c9efb9","premise_count":1,"theorem":"ray basis maps: for L f = (a + b t) f + kappa (t - h) f', b > 0, kappa >= 0, and g real-rooted with every root <= h, of multiplicity m at h, T(u^k) = L^k g sends every real polynomial with all roots in S to one with all roots in (-inf, h] if and only if S lies in (-inf, a + b h + kappa m] (RealRooted rayBasisMap_maps_realRootedIn_iff; the step is Ma and Wang, EJC 15 (2008) R17, Theorem 2)","lean_route":null,"lean_declarations":[],"certificate_hash":"20ccce3460e4d622026ff18b5df08c7e38ee1db2f5c2511d5a6518b9ceca1604","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-21d2f90cb223c8f5","role":"sequence","name":"shift_x_plus_one of Riordan array (1-x,x/(1-x)).","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]"}]},{"id":"e-479446aed5ce9d3c","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-5d685a1643c9efb9","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":"2ea078c728a63487a55ed0af69d1426e8d9aa81a1c0d4c7e46a7ceeae1ea5b29","verification_polytool":"replayed","verification_lean":"not_attempted","claims":[{"claim":"real_rooted","from_index":0,"region":{"upper":{"closed":true,"value":"0"}}},{"claim":"interlacing","from_index":1,"kind":"weak","orientation":"unordered"}],"premises":[{"position":0,"node_id":"n-9a6b0e2632270c41","role":"sequence","name":"basis_map_A008277 of Riordan array (1-x,x/(1-x)).","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:     4,       5,        1;\n  1:    12,      24,       10,        1;\n  2:    36,     108,       74,       16,        1;\n  3:   108,     468,      478,      170,       23,        1;\n  4:   324,    1980,     2858,     1498,      331,       31,       1;\n  5:   972,    8244,    16270,    11846,     3815,      579,      40,      1;\n  6:  2916,   33948,    89594,    87346,    38551,     8447,     939,     50,     1;\n  7:  8748,  138708,   481918,   613670,   357203,   106127,   16898,   1439,    61,    1;\n  8: 26244,  563580,  2548298,  4163938,  3114091,  1206219,  258209,  31288,  2110,   73,  1;\n  9: 78732, 2280564, 13305070, 27531926, 25962575, 12763843, 3530100, 571089, 54498, 2986, 86, 1;\n","recurrence_sentence":"The row polynomials satisfy P(n,x) = (3+x)*P(n-1,x) + x*d/dx P(n-1,x) for n >= 1, with P(0,x) = 4 + 5*x + x^2.","mathematica":"P[0, x_] := 4 + 5*x + x^2;\nP[n_, x_] := P[n, x] = Expand[(3 + x)*P[n - 1, x] + x*D[P[n - 1, x], x]];\nTable[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten","pari":"P(N) = {my(v = vector(N + 1)); v[1] = 4 + 5*x + x^2; for(n = 1, N, my(i = n + 1); v[i] = (3 + x)*v[i - 1] + x*deriv(v[i - 1], 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-7717563017cb9e5f","steps":["basis A008277","D+1"],"edges":["e-94a89e959b0863ff","e-479446aed5ce9d3c"],"text":"A159854 \u2192 basis A008277 \u2192 D+1"},"headline":"A159854 \u2192 basis A008277 \u2192 D+1","chain":null}