{"id":"n-f1b9721ef097c1eb","revision":0,"first_index":0,"fingerprint":"f6-9637c02c4490662e","name":"theta_plus_one_squared of basis_map_A013610 of Triangle read by rows: T(n,k) = binomial(2n+1, 2k+1).","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"],["4","12"],["16","144","81"],["64","960","1944","432"],["256","5376","19440","17280","2025"],["1024","27648","145152","241920","121500","8748"],["4096","135168","933120","2322432","2268000","734832","35721"],["16384","638976","5474304","18247680","27216000","17635968","4000752","139968"],["65536","2949120","30191616","126517248","256608000","258660864","120022560","20155392","531441"],["262144","13369344","159252480","805109760","2075673600","2882221056","2112397056","739031040","95659380","1968300"],["1048576","59768832","812187648","4812963840","15095808000","26900729856","27461161728","15371845632","4209012720","433026000","7144929"],["4194304","264241152","4034396160","27433893888","101523456000","221727227904","292919058432","230577684480","102138708672","22517352000","1886261256","25509168"],["16777216","1157627904","19619905536","150617456640","642981888000","1666609053696","2716158541824","2787428007936","1750949291520","630485856000","114433182864","7958860416","89813529"],["67108864","5033164800","93767860224","802236137472","3883106304000","11666263375872","22684401008640","28887890264064","23812910364672","12249439488000","3661861851648","557120229120","32692124556","312487308"],["268435456","21743271936","441660211200","4167460454400","22562891366400","77165088473088","174669887766528","266657448591360","274209270865920","186191480217600","80037837614592","20204893642752","2615369964480","131244669360","1076168025"]],"degrees":[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14],"row_sums":["1","16","241","3400","44377","545992","6433369","73370032","815701777","8884944160","95164697761","1005051983416","10488506263081","108337101769240","1109077621713097"],"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":[["0","0","0"],["0","-6","0"],["-16","84","-9"],["32","-216","54"],["0","144","-108"],["0","0","72"]]},"terms":[{"coeff":{"coeffs":[["0","0","108","-81"],["0","-24","432","378"],["-64","-624","-1368","-351"],["128","288","-1080","-1026"],["0","576","1728","540"],["0","0","288","648"]]},"deriv_order":0,"offset":1,"sign":"none"},{"coeff":{"coeffs":[["0","0","12","18","-54"],["0","0","72","288","216"],["0","-64","-336","-468","-108"],["0","0","-288","-1152","-864"],["0","0","0","-144","-432"]]},"deriv_order":1,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"}}],"properties":[{"id":184873,"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-974d7b6aea198c01"]},{"id":184874,"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-974d7b6aea198c01"]},{"id":184875,"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-974d7b6aea198c01"]},{"id":184876,"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-974d7b6aea198c01"]}],"region_exact":{},"incoming_edges":[{"id":"e-974d7b6aea198c01","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-f1b9721ef097c1eb","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":"a7c55a4310bcb5827766ed666a891d0ba1c11014be3034138b2830033877735b","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-966c4f25f43e716e","role":"sequence","name":"basis_map_A013610 of Triangle read by rows: T(n,k) = binomial(2n+1, 2k+1).","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:      4,       12;\n  2:     16,      144,        81;\n  3:     64,      960,      1944,       432;\n  4:    256,     5376,     19440,     17280,       2025;\n  5:   1024,    27648,    145152,    241920,     121500,       8748;\n  6:   4096,   135168,    933120,   2322432,    2268000,     734832,      35721;\n  7:  16384,   638976,   5474304,  18247680,   27216000,   17635968,    4000752,    139968;\n  8:  65536,  2949120,  30191616, 126517248,  256608000,  258660864,  120022560,  20155392,   531441;\n  9: 262144, 13369344, 159252480, 805109760, 2075673600, 2882221056, 2112397056, 739031040, 95659380, 1968300;\n","recurrence_sentence":"The row polynomials satisfy (32*n^3-16*n^2+(144*n^4-216*n^3+84*n^2-6*n)*x+(72*n^5-108*n^4+54*n^3-9*n^2)*x^2)*P(n,x) = (128*n^3-64*n^2+(576*n^4+288*n^3-624*n^2-24*n)*x+(288*n^5+1728*n^4-1080*n^3-1368*n^2+432*n+108)*x^2+(648*n^5+540*n^4-1026*n^3-351*n^2+378*n-81)*x^3)*P(n-1,x) - (64*n^2*x+(288*n^3+336*n^2-72*n-12)*x^2+(144*n^4+1152*n^3+468*n^2-288*n-18)*x^3+(432*n^4+864*n^3+108*n^2-216*n+54)*x^4)*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[((128*n^3 - 64*n^2 + (576*n^4 + 288*n^3 - 624*n^2 - 24*n)*x + (288*n^5 + 1728*n^4 - 1080*n^3 - 1368*n^2 + 432*n + 108)*x^2 + (648*n^5 + 540*n^4 - 1026*n^3 - 351*n^2 + 378*n - 81)*x^3)*P[n - 1, x] - (64*n^2*x + (288*n^3 + 336*n^2 - 72*n - 12)*x^2 + (144*n^4 + 1152*n^3 + 468*n^2 - 288*n - 18)*x^3 + (432*n^4 + 864*n^3 + 108*n^2 - 216*n + 54)*x^4)*D[P[n - 1, x], x])/(32*n^3 - 16*n^2 + (144*n^4 - 216*n^3 + 84*n^2 - 6*n)*x + (72*n^5 - 108*n^4 + 54*n^3 - 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] = ((128*n^3 - 64*n^2 + (576*n^4 + 288*n^3 - 624*n^2 - 24*n)*x + (288*n^5 + 1728*n^4 - 1080*n^3 - 1368*n^2 + 432*n + 108)*x^2 + (648*n^5 + 540*n^4 - 1026*n^3 - 351*n^2 + 378*n - 81)*x^3)*v[i - 1] - (64*n^2*x + (288*n^3 + 336*n^2 - 72*n - 12)*x^2 + (144*n^4 + 1152*n^3 + 468*n^2 - 288*n - 18)*x^3 + (432*n^4 + 864*n^3 + 108*n^2 - 216*n + 54)*x^4)*deriv(v[i - 1], x))/(32*n^3 - 16*n^2 + (144*n^4 - 216*n^3 + 84*n^2 - 6*n)*x + (72*n^5 - 108*n^4 + 54*n^3 - 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-c30cb06ad75c5e21","steps":["basis A013610","(\u03b8+1)\u00b2"],"edges":["e-71859d6a54438fd0","e-974d7b6aea198c01"],"text":"A103327 \u2192 basis A013610 \u2192 (\u03b8+1)\u00b2"},"headline":"A103327 \u2192 basis A013610 \u2192 (\u03b8+1)\u00b2","chain":null}