{"id":"n-57e795d3ee456994","revision":0,"first_index":0,"fingerprint":"f6-77376c12e17ddecb","name":"theta_plus_one_squared of basis_map_A013610 of Triangle whose (i,j)-th entry is binomial(i,j)*7^(i-j).","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"],["8","12"],["64","192","81"],["512","2304","1944","432"],["4096","24576","31104","13824","2025"],["32768","245760","414720","276480","81000","8748"],["262144","2359296","4976640","4423680","1944000","419904","35721"],["2097152","22020096","55738368","61931520","36288000","11757312","2000376","139968"],["16777216","201326592","594542592","792723456","580608000","250822656","64012032","8957952","531441"],["134217728","1811939328","6115295232","9512681472","8360755200","4514807808","1536288768","322486272","38263752","1968300"],["1073741824","16106127360","61152952320","108716359680","111476736000","72236924928","30725775360","8599633920","1530550080","157464000","7144929"],["8589934592","141733920768","597939978240","1195879956480","1401421824000","1059474898944","540773646336","189191946240","44896135680","6928416000","628753752","25509168"],["68719476736","1236950581248","5740223791104","12756052869120","16817061888000","14529941471232","8652378341376","3632485367808","1077507256320","221709312000","30180180096","2448880128","89813529"],["549755813888","10720238370816","54271206752256","132662949838848","194330492928000","188889239126016","128549621071872","62963079708672","22412150931456","5764442112000","1046246243328","127341766656","9340607016","312487308"],["4398046511104","92358976733184","506531263021056","1350750034722816","2176501520793600","2350621642457088","1799694695006208","1007409275338752","418360150720512","129123503308800","29294894813184","4754092621824","523073992896","34998578496","1076168025"]],"degrees":[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14],"row_sums":["1","20","337","5192","75625","1059476","14421385","191972792","2510301937","32348703860","411783410401","5187484920200","64765749228697","802296417758132","9870357244787545"],"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"]],"recurrence":{"denominator":{"coeffs":[["64","-24","0"],["0","72","0"],["0","0","9"]]},"terms":[{"coeff":{"coeffs":[["512","576","216","27"],["0","576","360","54"],["0","0","72","27"]]},"deriv_order":0,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"}}],"properties":[{"id":186027,"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-90e44d2a14fcdcaf"]},{"id":186028,"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-90e44d2a14fcdcaf"]},{"id":186029,"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-90e44d2a14fcdcaf"]},{"id":186030,"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-90e44d2a14fcdcaf"]}],"region_exact":{},"incoming_edges":[{"id":"e-90e44d2a14fcdcaf","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-57e795d3ee456994","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":"25ffe1613538a9f814e2232c10cf9eaae16b1e08577685a1700f092c0e6ce9f0","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-a227586ec36efe06","role":"sequence","name":"basis_map_A013610 of Triangle whose (i,j)-th entry is binomial(i,j)*7^(i-j).","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":[{"id":"l-bb6c52fbd3208c9d","kind":"fingerprint_related","node_a_id":"n-57e795d3ee456994","node_b_id":"n-8b2de826e1bf0a0d","fingerprint":"f6-77376c12e17ddecb","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-77376c12e17ddecb"},"other_id":"n-8b2de826e1bf0a0d","other":{"id":"n-8b2de826e1bf0a0d","name":"theta_plus_one_squared of Triangle whose (i,j)-th entry is binomial(i,j)*8^(i-j)*3^j.","first_index":0,"fingerprint":"f6-77376c12e17ddecb","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":0,"preview_rows":[["64","192","81"],["512","2304","1944","432"],["4096","24576","31104","13824","2025"]],"card_rows":[["64","192","81"],["512","2304","1944","432"],["4096","24576","31104","13824","2025"],["32768","245760","414720","276480","81000","8748"]],"row_sums_preview":["337","5192","75625","1059476","14421385","191972792","2510301937","32348703860"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-108b34f10b7468f4","steps":["(\u03b8+1)\u00b2"],"edges":["e-70abfa17c149d327"],"text":"A038281 \u2192 (\u03b8+1)\u00b2"},"headline":"A038281 \u2192 (\u03b8+1)\u00b2","chain":null}}],"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:         8,         12;\n  2:        64,        192,         81;\n  3:       512,       2304,       1944,        432;\n  4:      4096,      24576,      31104,      13824,       2025;\n  5:     32768,     245760,     414720,     276480,      81000,       8748;\n  6:    262144,    2359296,    4976640,    4423680,    1944000,     419904,      35721;\n  7:   2097152,   22020096,   55738368,   61931520,   36288000,   11757312,    2000376,    139968;\n  8:  16777216,  201326592,  594542592,  792723456,  580608000,  250822656,   64012032,   8957952,   531441;\n  9: 134217728, 1811939328, 6115295232, 9512681472, 8360755200, 4514807808, 1536288768, 322486272, 38263752, 1968300;\n","recurrence_sentence":"The row polynomials satisfy (64+(72*n-24)*x+9*n^2*x^2)*P(n,x) = (512+(576*n+576)*x+(72*n^2+360*n+216)*x^2+(27*n^2+54*n+27)*x^3)*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[((512 + (576*n + 576)*x + (72*n^2 + 360*n + 216)*x^2 + (27*n^2 + 54*n + 27)*x^3)*P[n - 1, x])/(64 + (72*n - 24)*x + 9*n^2*x^2)]];\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] = ((512 + (576*n + 576)*x + (72*n^2 + 360*n + 216)*x^2 + (27*n^2 + 54*n + 27)*x^3)*v[i - 1])/(64 + (72*n - 24)*x + 9*n^2*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-f211cb6153c37c6d","steps":["basis A013610","(\u03b8+1)\u00b2"],"edges":["e-892638ce5a781886","e-90e44d2a14fcdcaf"],"text":"A027466 \u2192 basis A013610 \u2192 (\u03b8+1)\u00b2"},"headline":"A027466 \u2192 basis A013610 \u2192 (\u03b8+1)\u00b2","chain":null}
