{"id":"n-314a54b7f65b20a6","revision":0,"first_index":0,"fingerprint":"f6-e20b2a4d6e4d3436","name":"theta_plus_one_squared of Triangle of coefficients in expansion of (1+11x)^n.","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":[["1","88","1089"],["1","132","3267","21296"],["1","176","6534","85184","366025"],["1","220","10890","212960","1830125","5797836"],["1","264","16335","425920","5490375","34787016","86806489"],["1","308","22869","745360","12810875","121754556","607645423","1247178944"],["1","352","30492","1192576","25621750","324678816","2430581692","9977431552","17363069361"],["1","396","39204","1788864","46119150","730527336","7291745076","44898441984","156267624249","235794769100"],["1","440","49005","2555520","76865250","1461054672","18229362690","149661473280","781338121245","2357947691000","3138428376721"],["1","484","59895","3513840","120788250","2678600232","40104597918","411569051520","2864906444565","12968712300500","34522712143931","41084880567984"],["1","528","71874","4685120","181182375","4591886112","80209195836","987765723648","8594719333695","51874849202000","207136272863586","493018566815808","530394395665849"],["1","572","84942","6090656","261707875","7461814932","148959935124","2140159067904","22346270267607","168593259906500","897590515742206","3204620684302752","6895127143656037","6766451580210476"],["1","616","99099","7751744","366391025","11607267672","260679886467","4280318135808","52141297291083","472061127738200","3141566805097721","14954896526746176","48265890005592259","94730322122946664","85443712556229225"],["1","660","114345","9689680","499624125","17410901508","434466477445","8025596504640","111731351338035","1180152819345500","9424700415293163","56080861975298160","241329450027961295","710477415922099980","1281655688343438375","1069375531370406656"],["1","704","130680","11925760","666165500","25324947648","695146363912","14267727119360","223462702676070","2697492158504000","25132534440781768","179458758320954112","965317800111845180","3789212884917866560","10253245506747507000","17110008501926506496","13279471930572354529"]],"degrees":[2,3,4,5,6,7,8,9,10,11,12,13,14,15,16],"row_sums":["1178","24696","457920","7852032","127526400","1990158336","30122606592","445031055360","6447145549824","91895688069120","1292091556626432","17957026302787584","247065143421173760","3369644010208493568","45604783860775649280"],"nonn":true,"definitions":[{"position":0,"source":"derived:theta_plus_one_squared","source_kind":"derived","equality_edge_id":null,"recurrence":{"first_index":0,"initial_polynomials":[["1","88","1089"]],"recurrence":{"denominator":{"coeffs":[["1","55","484"],["0","33","484"],["0","0","121"]]},"terms":[{"coeff":{"coeffs":[["1","99","2057","11979"],["0","33","1089","7986"],["0","0","121","1331"]]},"deriv_order":0,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"}}],"properties":[{"id":191487,"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-51d34a8717193b90"]},{"id":191488,"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-51d34a8717193b90"]},{"id":191489,"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-51d34a8717193b90"]},{"id":191490,"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-51d34a8717193b90"]}],"region_exact":{},"incoming_edges":[{"id":"e-51d34a8717193b90","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-314a54b7f65b20a6","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":"bf7e94765f059ce00cf0207ff1f9ebfb3b687dd5d0ecb4d4e3d1274b67a595f2","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-b1e393c17b0898c2","role":"sequence","name":"Triangle of coefficients in expansion of (1+11x)^n.","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,  88,  1089;\n  1: 1, 132,  3267,   21296;\n  2: 1, 176,  6534,   85184,    366025;\n  3: 1, 220, 10890,  212960,   1830125,    5797836;\n  4: 1, 264, 16335,  425920,   5490375,   34787016,    86806489;\n  5: 1, 308, 22869,  745360,  12810875,  121754556,   607645423,   1247178944;\n  6: 1, 352, 30492, 1192576,  25621750,  324678816,  2430581692,   9977431552,   17363069361;\n  7: 1, 396, 39204, 1788864,  46119150,  730527336,  7291745076,  44898441984,  156267624249,   235794769100;\n  8: 1, 440, 49005, 2555520,  76865250, 1461054672, 18229362690, 149661473280,  781338121245,  2357947691000,  3138428376721;\n  9: 1, 484, 59895, 3513840, 120788250, 2678600232, 40104597918, 411569051520, 2864906444565, 12968712300500, 34522712143931, 41084880567984;\n","recurrence_sentence":"The row polynomials satisfy (1+(33*n+55)*x+(121*n^2+484*n+484)*x^2)*P(n,x) = (1+(33*n+99)*x+(121*n^2+1089*n+2057)*x^2+(1331*n^2+7986*n+11979)*x^3)*P(n-1,x) for n >= 1, with P(0,x) = 1 + 88*x + 1089*x^2.","mathematica":"P[0, x_] := 1 + 88*x + 1089*x^2;\nP[n_, x_] := P[n, x] = Expand[Together[((1 + (33*n + 99)*x + (121*n^2 + 1089*n + 2057)*x^2 + (1331*n^2 + 7986*n + 11979)*x^3)*P[n - 1, x])/(1 + (33*n + 55)*x + (121*n^2 + 484*n + 484)*x^2)]];\nTable[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten","pari":"P(N) = {my(v = vector(N + 1)); v[1] = 1 + 88*x + 1089*x^2; for(n = 1, N, my(i = n + 1); v[i] = ((1 + (33*n + 99)*x + (121*n^2 + 1089*n + 2057)*x^2 + (1331*n^2 + 7986*n + 11979)*x^3)*v[i - 1])/(1 + (33*n + 55)*x + (121*n^2 + 484*n + 484)*x^2)); 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-b1e393c17b0898c2","steps":["(\u03b8+1)\u00b2"],"edges":["e-51d34a8717193b90"],"text":"A013618 \u2192 (\u03b8+1)\u00b2"},"headline":"A013618 \u2192 (\u03b8+1)\u00b2","chain":null}