{"id":"n-352c7957093ee52b","revision":0,"first_index":1,"fingerprint":"f6-dc39590f11d4c375","name":"theta_plus_one of Triangle T(n, k) read by rows: row n gives the coefficients of the numerator polynomials of the o.g.f. of the (n+1)-th diagonal of the Sheffer triangle A154537 (S2[2,1] generalized Stirling2), for n >= 0.","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"],["1","4"],["1","32","36"],["1","132","852","480"],["1","448","8616","23808","8400"],["1","1412","63240","465344","730320","181440"],["1","4320","396684","6124288","24023280","24781824","4656960"],["1","13060","2281500","65084320","507113200","1252749888","927904320","138378240"],["1","39296","12465360","607690240","8306367200","39338754048","67601334528","38131384320","4670265600"],["1","118020","65981328","5217819520","115776396000","931108326528","2989997149440","3816834822144","1709511194880","176432256000"],["1","354208","342298740","42331093504","1446368576800","18300228753408","97529787513984","228109502300160","226582892358912","83129401958400","7374868300800"],["1","1062788","1752936036","330091130208","16721287872800","315854907570048","2591603784178304","9886474164000768","17709561434836224","14167397101593600","4361219973411840","337903056691200"],["1","3188544","8901779832","2502292251392","182675476050480","4959254346375168","59485666645871872","343778083609460736","990644117331269376","1410463392953057280","933129007276308480","245656496126361600","16838835658444800"],["1","9565828","44958483864","18584259339584","1913075701478960","72527064653494080","1224605811916474624","10187459061907075072","43913474078876612352","99446039737067842560","115789312552200984576","64691716598570926080","14791875914853273600","906706535454720000"],["1","28697696","226255991004","135964390601984","19401331785711760","1004409886948162560","23196757708481409472","267574944873921462272","1640478592023248247552","5500138179267477135360","10088920623350444553216","9824739985525602189312","4714923862742012375040","948404402831189606400","52459449551308800000"]],"degrees":[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14],"row_sums":["1","5","69","1465","41273","1441757","59987357","2893524529","158668300593","9744944063861","662515723508917","49387067555283817","4005140040620419561","351025649060430271181","33061860745283094943629"],"nonn":true,"definitions":[{"position":0,"source":"derived:theta_plus_one","source_kind":"derived","equality_edge_id":null,"recurrence":{"first_index":1,"initial_polynomials":[["1"],["1","4"]],"recurrence":{"denominator":{"coeffs":[["-9"],["4"]]},"terms":[{"coeff":{"coeffs":[["-5","220","1980"],["4","-256","-3184"],["0","64","1600"],["0","0","-256"]]},"deriv_order":0,"offset":2,"sign":"none"},{"coeff":{"coeffs":[["0","-10","-160","360"],["0","8","208","-448"],["0","0","-64","128"]]},"deriv_order":1,"offset":2,"sign":"none"},{"coeff":{"coeffs":[["-4","252"],["0","-256"],["0","64"]]},"deriv_order":0,"offset":1,"sign":"none"},{"coeff":{"coeffs":[["0","-18","36"],["0","8","-16"]]},"deriv_order":1,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"}}],"properties":[{"id":191177,"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-994728aa03c15e27"]},{"id":191178,"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-994728aa03c15e27"]},{"id":191179,"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-994728aa03c15e27"]},{"id":191180,"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-994728aa03c15e27"]}],"region_exact":{},"incoming_edges":[{"id":"e-994728aa03c15e27","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-352c7957093ee52b","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":"5615e631412347df821a9e1c11dd2b1f89607c732e51650528bb0d1a2195316b","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-945ed98594b7abe2","role":"sequence","name":"Triangle T(n, k) read by rows: row n gives the coefficients of the numerator polynomials of the o.g.f. of the (n+1)-th diagonal of the Sheffer triangle A154537 (S2[2,1] generalized Stirling2), for n >= 0.","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: 1,      4;\n   3: 1,     32,       36;\n   4: 1,    132,      852,        480;\n   5: 1,    448,     8616,      23808,         8400;\n   6: 1,   1412,    63240,     465344,       730320,       181440;\n   7: 1,   4320,   396684,    6124288,     24023280,     24781824,       4656960;\n   8: 1,  13060,  2281500,   65084320,    507113200,   1252749888,     927904320,     138378240;\n   9: 1,  39296, 12465360,  607690240,   8306367200,  39338754048,   67601334528,   38131384320,    4670265600;\n  10: 1, 118020, 65981328, 5217819520, 115776396000, 931108326528, 2989997149440, 3816834822144, 1709511194880, 176432256000;\n","recurrence_sentence":"The row polynomials satisfy (4*n-9)*P(n,x) = (4*n-5+(64*n^2-256*n+220)*x-(256*n^3-1600*n^2+3184*n-1980)*x^2)*P(n-2,x) + ((8*n-10)*x-(64*n^2-208*n+160)*x^2+(128*n^2-448*n+360)*x^3)*d/dx P(n-2,x) - (4-(64*n^2-256*n+252)*x)*P(n-1,x) + ((8*n-18)*x-(16*n-36)*x^2)*d/dx P(n-1,x) for n >= 3, with P(1,x) = 1, P(2,x) = 1 + 4*x.","mathematica":"P[1, x_] := 1;\nP[2, x_] := 1 + 4*x;\nP[n_, x_] := P[n, x] = Expand[Together[((4*n - 5 + (64*n^2 - 256*n + 220)*x - (256*n^3 - 1600*n^2 + 3184*n - 1980)*x^2)*P[n - 2, x] + ((8*n - 10)*x - (64*n^2 - 208*n + 160)*x^2 + (128*n^2 - 448*n + 360)*x^3)*D[P[n - 2, x], x] - (4 - (64*n^2 - 256*n + 252)*x)*P[n - 1, x] + ((8*n - 18)*x - (16*n - 36)*x^2)*D[P[n - 1, x], x])/(4*n - 9)]];\nTable[CoefficientList[P[n, x], x], {n, 1, 10}] // Flatten","pari":"P(N) = {my(v = vector(N)); v[1] = 1; v[2] = 1 + 4*x; for(n = 3, N, my(i = n); v[i] = ((4*n - 5 + (64*n^2 - 256*n + 220)*x - (256*n^3 - 1600*n^2 + 3184*n - 1980)*x^2)*v[i - 2] + ((8*n - 10)*x - (64*n^2 - 208*n + 160)*x^2 + (128*n^2 - 448*n + 360)*x^3)*deriv(v[i - 2], x) - (4 - (64*n^2 - 256*n + 252)*x)*v[i - 1] + ((8*n - 18)*x - (16*n - 36)*x^2)*deriv(v[i - 1], x))/(4*n - 9)); 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-945ed98594b7abe2","steps":["\u03b8+1"],"edges":["e-994728aa03c15e27"],"text":"A290315 \u2192 \u03b8+1"},"headline":"A290315 \u2192 \u03b8+1","chain":null}