{"id":"n-99236eea6c591277","revision":0,"first_index":0,"fingerprint":"f6-eb0da916da09bf7a","name":"theta_plus_one of basis_map_A013611 of Triangle read by rows: T(n,k) = binomial(k,n-k).","polytool_family":null,"interpretation":null,"rows_observed":15,"degree_pattern":"affine","degree_slope":1,"degree_intercept":0,"palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":false,"rows":[["1"],["2","8"],["3","32","48"],["5","80","288","256"],["8","176","1008","2048","1280"],["13","360","2832","9216","12800","6144"],["21","704","7056","31744","70400","73728","28672"],["34","1336","16272","93696","288000","479232","401408","131072"],["55","2480","35568","249856","985600","2273280","3010560","2097152","589824"],["89","4528","74736","620032","2991360","8865792","16257024","17825792","10616832","2621440"],["144","8160","152352","1458176","8326400","30228480","71249920","108003328","100859904","52428800","11534336"],["233","14552","303216","3289344","21708800","93419520","269946880","525860864","677707776","550502400","253755392","50331648"],["377","25728","591840","7177216","53772800","267816960","918220800","2193620992","3630956544","4063232000","2918187008","1207959552","218103808"],["610","45160","1136736","15240192","127814400","723548160","2873937920","8151105536","16546332672","23763353600","23472373760","15099494400","5670699008","939524096"],["987","78784","2153904","31636480","293674240","1862983680","8418529280","27680309248","66728558592","117503426560","148827537408","131466264576","76554436608","26306674688","4026531840"]],"degrees":[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14],"row_sums":["1","10","83","629","4520","31365","212325","1411050","9244375","59877625","384250000","2446840625","15479665625","97384606250","609702796875"],"nonn":true,"definitions":[{"position":0,"source":"derived:theta_plus_one","source_kind":"derived","equality_edge_id":null,"recurrence":{"first_index":0,"initial_polynomials":[["1"]],"recurrence":{"denominator":{"coeffs":[["40","128","0","0"],["40","208","64","-256"],["0","80","128","-256"],["0","0","64","0"]]},"terms":[{"coeff":{"coeffs":[["240","1088","1408","-1024","-2048"],["-80","624","2304","768","-2048"],["0","-240","256","1280","0"],["0","0","-128","0","0"]]},"deriv_order":0,"offset":1,"sign":"none"},{"coeff":{"coeffs":[["25","180","288","-512","-1792","-1024"],["0","100","560","384","-1280","-1024"],["0","0","80","384","256","0"]]},"deriv_order":1,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"}}],"properties":[{"id":179752,"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-551137b06775de04"]},{"id":179753,"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-551137b06775de04"]},{"id":179754,"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-551137b06775de04"]}],"region_exact":{},"incoming_edges":[{"id":"e-551137b06775de04","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-99236eea6c591277","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":"bad9b7a50765872a9bcec76c909dac7dba5e383cc31550af381a1eee8dfbb5be","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-0552ada0abe44307","role":"sequence","name":"basis_map_A013611 of Triangle read by rows: T(n,k) = binomial(k,n-k).","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:  1;\n  1:  2,    8;\n  2:  3,   32,    48;\n  3:  5,   80,   288,    256;\n  4:  8,  176,  1008,   2048,    1280;\n  5: 13,  360,  2832,   9216,   12800,    6144;\n  6: 21,  704,  7056,  31744,   70400,   73728,    28672;\n  7: 34, 1336, 16272,  93696,  288000,  479232,   401408,   131072;\n  8: 55, 2480, 35568, 249856,  985600, 2273280,  3010560,  2097152,   589824;\n  9: 89, 4528, 74736, 620032, 2991360, 8865792, 16257024, 17825792, 10616832, 2621440;\n","recurrence_sentence":"The row polynomials satisfy (40*n+40+(80*n^2+208*n+128)*x+(64*n^3+128*n^2+64*n)*x^2-(256*n^2+256*n)*x^3)*P(n,x) = -(80*n-240+(240*n^2-624*n-1088)*x+(128*n^3-256*n^2-2304*n-1408)*x^2-(1280*n^2+768*n-1024)*x^3+(2048*n+2048)*x^4)*P(n-1,x) + (25+(100*n+180)*x+(80*n^2+560*n+288)*x^2+(384*n^2+384*n-512)*x^3+(256*n^2-1280*n-1792)*x^4-(1024*n+1024)*x^5)*d/dx P(n-1,x) for n >= 1, with P(0,x) = 1.","mathematica":"P[0, x_] := 1;\nP[n_, x_] := P[n, x] = Expand[Together[(-(80*n - 240 + (240*n^2 - 624*n - 1088)*x + (128*n^3 - 256*n^2 - 2304*n - 1408)*x^2 - (1280*n^2 + 768*n - 1024)*x^3 + (2048*n + 2048)*x^4)*P[n - 1, x] + (25 + (100*n + 180)*x + (80*n^2 + 560*n + 288)*x^2 + (384*n^2 + 384*n - 512)*x^3 + (256*n^2 - 1280*n - 1792)*x^4 - (1024*n + 1024)*x^5)*D[P[n - 1, x], x])/(40*n + 40 + (80*n^2 + 208*n + 128)*x + (64*n^3 + 128*n^2 + 64*n)*x^2 - (256*n^2 + 256*n)*x^3)]];\nTable[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten","pari":"P(N) = {my(v = vector(N + 1)); v[1] = 1; for(n = 1, N, my(i = n + 1); v[i] = (-(80*n - 240 + (240*n^2 - 624*n - 1088)*x + (128*n^3 - 256*n^2 - 2304*n - 1408)*x^2 - (1280*n^2 + 768*n - 1024)*x^3 + (2048*n + 2048)*x^4)*v[i - 1] + (25 + (100*n + 180)*x + (80*n^2 + 560*n + 288)*x^2 + (384*n^2 + 384*n - 512)*x^3 + (256*n^2 - 1280*n - 1792)*x^4 - (1024*n + 1024)*x^5)*deriv(v[i - 1], x))/(40*n + 40 + (80*n^2 + 208*n + 128)*x + (64*n^3 + 128*n^2 + 64*n)*x^2 - (256*n^2 + 256*n)*x^3)); 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-21854f4257effb01","steps":["basis A013611","\u03b8+1"],"edges":["e-3723cb941a96ee35","e-551137b06775de04"],"text":"A030528 \u2192 basis A013611 \u2192 \u03b8+1"},"headline":"A030528 \u2192 basis A013611 \u2192 \u03b8+1","chain":null}