{"id":"n-8bc628d590897076","revision":0,"first_index":1,"fingerprint":"f6-84e29d07816d0d9d","name":"theta_plus_one_squared of basis_map_A013610 of Square array of binomial coefficients T(n,k) = binomial(n,k), n >= 0, k >= 0, read by downward antidiagonals.","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","12"],["2","36","81"],["3","84","405","432"],["5","180","1296","3024","2025"],["8","360","3483","12528","18225","8748"],["13","696","8424","41040","93150","96228","35721"],["21","1308","19035","117072","362475","586116","464373","139968"],["34","2412","40986","304560","1196775","2650644","3286332","2099520","531441"],["55","4380","85131","741744","3535650","9972720","16967475","16936128","9034497","1968300"],["89","7860","171963","1718928","9636975","33067440","71799210","98397504","81841914","37397700","7144929"],["144","13968","339714","3831840","24707025","99945900","264514005","463154112","528783795","375945300","150043509","25509168"],["233","24624","659016","8279712","60363225","281379420","879451020","1879350336","2742767001","2674919700","1657623528","586710864","89813529"],["377","43116","1259226","17438112","141843150","748828800","2701043415","6824979648","12165747372","15167719800","12882306987","7066039536","2245338225","312487308"],["610","75060","2375487","35948448","322758675","1903739760","7787178000","22734022464","47957767281","73059359400","79308711900","59563907280","29279210454","8437157316","1076168025"]],"degrees":[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14],"row_sums":["1","13","119","924","6530","43352","275272","1690368","10112704","59246080","341184512","1936788480","10861342208","60275075072","331468380160"],"nonn":true,"definitions":[{"position":0,"source":"derived:theta_plus_one_squared","source_kind":"derived","equality_edge_id":null,"recurrence":{"first_index":1,"initial_polynomials":[["1"],["1","12"]],"recurrence":{"denominator":{"coeffs":[["25","-255","306","702","81","-243"],["0","225","-990","-486","1134","243"],["0","0","540","-675","-1053","243"],["0","0","0","405","0","-243"],["0","0","0","0","81","0"]]},"terms":[{"coeff":{"coeffs":[["25","45","-234","-486","81","729","0"],["0","225","765","-351","-2511","-1944","0"],["0","0","540","2160","1053","-2187","-1458"],["0","0","0","405","1539","729","-729"],["0","0","0","0","81","243","0"]]},"deriv_order":0,"offset":2,"sign":"none"},{"coeff":{"coeffs":[["25","45","-234","-486","81","729","0"],["0","225","450","-1836","-3888","-1215","0"],["0","0","540","1755","-486","-2916","-729"],["0","0","0","405","1377","243","-729"],["0","0","0","0","81","243","0"]]},"deriv_order":0,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"}}],"properties":[{"id":178120,"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-6aa0ee9ff3d97163"]},{"id":178121,"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-6aa0ee9ff3d97163"]},{"id":178122,"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-6aa0ee9ff3d97163"]}],"region_exact":{},"incoming_edges":[{"id":"e-6aa0ee9ff3d97163","operation":{"index_shift":0,"operation":"preserver_image","operator":"{\"shape\":\"euler_polynomial\",\"coefficients\":[{\"numerator\":[\"1\"],\"denominator\":[\"1\"]},{\"numerator\":[\"2\"],\"denominator\":[\"1\"]},{\"numerator\":[\"1\"],\"denominator\":[\"1\"]}]}"},"operation_detail":"euler_polynomial, shift 0","conclusion_id":"n-8bc628d590897076","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":"335032b6c756664058341c72c5705d8a1985ad7686f5c282e93204fada63a7be","verification_polytool":"replayed","verification_lean":"not_attempted","claims":[{"claim":"real_rooted","from_index":1,"region":{"upper":{"closed":true,"value":"0"}}}],"premises":[{"position":0,"node_id":"n-7f247edfefc78f8c","role":"sequence","name":"basis_map_A013610 of Square array of binomial coefficients T(n,k) = binomial(n,k), n >= 0, k >= 0, read by downward antidiagonals.","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,   12;\n   3:  2,   36,    81;\n   4:  3,   84,   405,    432;\n   5:  5,  180,  1296,   3024,    2025;\n   6:  8,  360,  3483,  12528,   18225,    8748;\n   7: 13,  696,  8424,  41040,   93150,   96228,    35721;\n   8: 21, 1308, 19035, 117072,  362475,  586116,   464373,   139968;\n   9: 34, 2412, 40986, 304560, 1196775, 2650644,  3286332,  2099520,  531441;\n  10: 55, 4380, 85131, 741744, 3535650, 9972720, 16967475, 16936128, 9034497, 1968300;\n","recurrence_sentence":"The row polynomials satisfy (25+(225*n-255)*x+(540*n^2-990*n+306)*x^2+(405*n^3-675*n^2-486*n+702)*x^3+(81*n^4-1053*n^2+1134*n+81)*x^4-(243*n^3-243*n^2-243*n+243)*x^5)*P(n,x) = (25+(225*n+45)*x+(540*n^2+765*n-234)*x^2+(405*n^3+2160*n^2-351*n-486)*x^3+(81*n^4+1539*n^3+1053*n^2-2511*n+81)*x^4+(243*n^4+729*n^3-2187*n^2-1944*n+729)*x^5-(729*n^3+1458*n^2)*x^6)*P(n-2,x) + (25+(225*n+45)*x+(540*n^2+450*n-234)*x^2+(405*n^3+1755*n^2-1836*n-486)*x^3+(81*n^4+1377*n^3-486*n^2-3888*n+81)*x^4+(243*n^4+243*n^3-2916*n^2-1215*n+729)*x^5-(729*n^3+729*n^2)*x^6)*P(n-1,x) for n >= 3, with P(1,x) = 1, P(2,x) = 1 + 12*x.","mathematica":"P[1, x_] := 1;\nP[2, x_] := 1 + 12*x;\nP[n_, x_] := P[n, x] = Expand[Together[((25 + (225*n + 45)*x + (540*n^2 + 765*n - 234)*x^2 + (405*n^3 + 2160*n^2 - 351*n - 486)*x^3 + (81*n^4 + 1539*n^3 + 1053*n^2 - 2511*n + 81)*x^4 + (243*n^4 + 729*n^3 - 2187*n^2 - 1944*n + 729)*x^5 - (729*n^3 + 1458*n^2)*x^6)*P[n - 2, x] + (25 + (225*n + 45)*x + (540*n^2 + 450*n - 234)*x^2 + (405*n^3 + 1755*n^2 - 1836*n - 486)*x^3 + (81*n^4 + 1377*n^3 - 486*n^2 - 3888*n + 81)*x^4 + (243*n^4 + 243*n^3 - 2916*n^2 - 1215*n + 729)*x^5 - (729*n^3 + 729*n^2)*x^6)*P[n - 1, x])/(25 + (225*n - 255)*x + (540*n^2 - 990*n + 306)*x^2 + (405*n^3 - 675*n^2 - 486*n + 702)*x^3 + (81*n^4 - 1053*n^2 + 1134*n + 81)*x^4 - (243*n^3 - 243*n^2 - 243*n + 243)*x^5)]];\nTable[CoefficientList[P[n, x], x], {n, 1, 10}] // Flatten","pari":"P(N) = {my(v = vector(N)); v[1] = 1; v[2] = 1 + 12*x; for(n = 3, N, my(i = n); v[i] = ((25 + (225*n + 45)*x + (540*n^2 + 765*n - 234)*x^2 + (405*n^3 + 2160*n^2 - 351*n - 486)*x^3 + (81*n^4 + 1539*n^3 + 1053*n^2 - 2511*n + 81)*x^4 + (243*n^4 + 729*n^3 - 2187*n^2 - 1944*n + 729)*x^5 - (729*n^3 + 1458*n^2)*x^6)*v[i - 2] + (25 + (225*n + 45)*x + (540*n^2 + 450*n - 234)*x^2 + (405*n^3 + 1755*n^2 - 1836*n - 486)*x^3 + (81*n^4 + 1377*n^3 - 486*n^2 - 3888*n + 81)*x^4 + (243*n^4 + 243*n^3 - 2916*n^2 - 1215*n + 729)*x^5 - (729*n^3 + 729*n^2)*x^6)*v[i - 1])/(25 + (225*n - 255)*x + (540*n^2 - 990*n + 306)*x^2 + (405*n^3 - 675*n^2 - 486*n + 702)*x^3 + (81*n^4 - 1053*n^2 + 1134*n + 81)*x^4 - (243*n^3 - 243*n^2 - 243*n + 243)*x^5)); 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-392f49b496f79fbf","steps":["basis A013610","(\u03b8+1)\u00b2"],"edges":["e-74fcd2d5aef5a8cd","e-6aa0ee9ff3d97163"],"text":"A026729 \u2192 basis A013610 \u2192 (\u03b8+1)\u00b2"},"headline":"A026729 \u2192 basis A013610 \u2192 (\u03b8+1)\u00b2","chain":null}