{"id":"n-106a2e627392653c","revision":0,"first_index":0,"fingerprint":"f6-2886ec3559ef5b58","name":"theta_plus_one of Triangle of unsigned 2-Lah numbers.","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":[["20","20","3"],["120","180","54","4"],["840","1680","756","112","5"],["6720","16800","10080","2240","200","6"],["60480","181440","136080","40320","5400","324","7"],["604800","2116800","1905120","705600","126000","11340","490","8"],["6652800","26611200","27941760","12418560","2772000","332640","21560","704","9"],["79833600","359251200","431101440","223534080","59875200","8981280","776160","38016","972","10"],["1037836800","5189184000","7005398400","4151347200","1297296000","233513280","25225200","1647360","63180","1300","11"],["14529715200","79913433600","119870150400","79913433600","28540512000","5993507520","776936160","63423360","3243240","100100","1694","12"],["217945728000","1307674368000","2157662707200","1598268672000","642161520000","154118764800","23308084800","2283240960","145945800","6006000","152460","2160","13"],["3487131648000","22666355712000","40799440281600","33243988377600","14841066240000","4007087884800","692583091200","79152353280","6071345280","312312000","10570560","224640","2704","14"],["59281238016000","414968666112000","809188898918400","719279021260800","353217376512000","105965212953600","20604346963200","2691180011520","240830029440","14866051200","628948320","17821440","321776","3332","15"],["1067062284288000","8002967132160000","16806230977536000","16183777978368000","8669881059840000","2861060749747200","618130408896000","90827325388800","9289158278400","668972304000","33963209280","1202947200","28959840","449820","4050","16"],["20274183401472000","162193467211776000","364935301226496000","378451423494144000","219636986849280000","79069315265740800","18791164430438400","3067945213132800","352988014579200","29052511488000","1720802603520","73139189760","2200947840","45581760","615600","4864","17"]],"degrees":[2,3,4,5,6,7,8,9,10,11,12,13,14,15,16],"row_sums":["43","358","3393","36046","424051","5470158","76751233","1163391958","18941512731","329604456886","6103575192193","119823200043678","2485452283923043","54309931242376606","1246803623807490561"],"nonn":true,"definitions":[{"position":0,"source":"derived:theta_plus_one","source_kind":"derived","equality_edge_id":null,"recurrence":{"first_index":0,"initial_polynomials":[["20","20","3"]],"recurrence":{"denominator":{"coeffs":[["2","2"],["0","1"]]},"terms":[{"coeff":{"coeffs":[["10","14","3"],["2","7","1"],["0","1","0"]]},"deriv_order":0,"offset":1,"sign":"none"},{"coeff":{"coeffs":[["0","2","3"],["0","0","1"]]},"deriv_order":1,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"}}],"properties":[{"id":189970,"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-c17e9ddfa12ea8f1"]},{"id":189971,"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-c17e9ddfa12ea8f1"]},{"id":189972,"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-c17e9ddfa12ea8f1"]}],"region_exact":{},"incoming_edges":[{"id":"e-c17e9ddfa12ea8f1","operation":{"index_shift":0,"operation":"preserver_image","operator":"{\"shape\":\"euler_polynomial\",\"coefficients\":[{\"numerator\":[\"1\"],\"denominator\":[\"1\"]},{\"numerator\":[\"1\"],\"denominator\":[\"1\"]}]}"},"operation_detail":"euler_polynomial, shift 0","conclusion_id":"n-106a2e627392653c","premise_count":1,"theorem":"Laguerre's theorem on multiplier sequences: if f is a real polynomial whose roots are all real and <= 0, then (f(k))_(k >= 0) is a multiplier sequence, so x^k -> f(k) x^k maps every real-rooted polynomial to a real-rooted polynomial or zero (Laguerre; see T. Craven and G. Csordas, Multiplier sequences for fields, Illinois J. Math. 21 (1977); Polya--Schur, Theorem 40 of P. Branden, arXiv:1410.6601: sum_k f(k) x^k / k! = f(theta) e^x = C e^x x^m prod (1 + x/a_j), a_j > 0); f(k) has one sign for k >= 0, so coefficients of one sign stay of one sign; applied to f(n, .) for each n","lean_route":null,"lean_declarations":["RealRooted.isMultiplierSequence_eval_of_roots_nonpos","RealRooted.isPFMultiplierSequence_eval_of_roots_nonpos","RealRooted.isMultiplierSequence_iff_preserves"],"certificate_hash":"a98c10ef89bb8c8c515693397799c8824e41cf717fef03fdd259ad6f76036252","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-f3352d3240f9f08b","role":"sequence","name":"Triangle of unsigned 2-Lah numbers.","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:          20,          20,            3;\n  1:         120,         180,           54,           4;\n  2:         840,        1680,          756,         112,           5;\n  3:        6720,       16800,        10080,        2240,         200,          6;\n  4:       60480,      181440,       136080,       40320,        5400,        324,         7;\n  5:      604800,     2116800,      1905120,      705600,      126000,      11340,       490,        8;\n  6:     6652800,    26611200,     27941760,    12418560,     2772000,     332640,     21560,      704,       9;\n  7:    79833600,   359251200,    431101440,   223534080,    59875200,    8981280,    776160,    38016,     972,     10;\n  8:  1037836800,  5189184000,   7005398400,  4151347200,  1297296000,  233513280,  25225200,  1647360,   63180,   1300,   11;\n  9: 14529715200, 79913433600, 119870150400, 79913433600, 28540512000, 5993507520, 776936160, 63423360, 3243240, 100100, 1694, 12;\n","recurrence_sentence":"The row polynomials satisfy (2+(n+2)*x)*P(n,x) = (2*n+10+(n^2+7*n+14)*x+(n+3)*x^2)*P(n-1,x) + (2*x+(n+3)*x^2)*d/dx P(n-1,x) for n >= 1, with P(0,x) = 20 + 20*x + 3*x^2.","mathematica":"P[0, x_] := 20 + 20*x + 3*x^2;\nP[n_, x_] := P[n, x] = Expand[Together[((2*n + 10 + (n^2 + 7*n + 14)*x + (n + 3)*x^2)*P[n - 1, x] + (2*x + (n + 3)*x^2)*D[P[n - 1, x], x])/(2 + (n + 2)*x)]];\nTable[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten","pari":"P(N) = {my(v = vector(N + 1)); v[1] = 20 + 20*x + 3*x^2; for(n = 1, N, my(i = n + 1); v[i] = ((2*n + 10 + (n^2 + 7*n + 14)*x + (n + 3)*x^2)*v[i - 1] + (2*x + (n + 3)*x^2)*deriv(v[i - 1], x))/(2 + (n + 2)*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-f3352d3240f9f08b","steps":["\u03b8+1"],"edges":["e-c17e9ddfa12ea8f1"],"text":"A143497 \u2192 \u03b8+1"},"headline":"A143497 \u2192 \u03b8+1","chain":null}