{"id":"n-d99f7b722c4c442c","revision":0,"first_index":0,"fingerprint":"f6-23cc3a53c9ba374d","name":"theta_plus_one_squared of basis_map_A013611 of Triangle of coefficients in expansion of (3+5x)^n.","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","80"],["64","1280","3600"],["512","15360","86400","128000"],["4096","163840","1382400","4096000","4000000"],["32768","1638400","18432000","81920000","160000000","115200000"],["262144","15728640","221184000","1310720000","3840000000","5529600000","3136000000"],["2097152","146800640","2477260800","18350080000","71680000000","154828800000","175616000000","81920000000"],["16777216","1342177280","26424115200","234881024000","1146880000000","3303014400000","5619712000000","5242880000000","2073600000000"],["134217728","12079595520","271790899200","2818572288000","16515072000000","59454259200000","134873088000000","188743680000000","149299200000000","51200000000000"],["1073741824","107374182400","2717908992000","32212254720000","220200960000000","951268147200000","2697461760000000","5033164800000000","5971968000000000","4096000000000000","1239040000000000"],["8589934592","944892805120","26575110144000","354334801920000","2768240640000000","13951932825600000","47475326976000000","110729625600000000","175177728000000000","180224000000000000","109035520000000000","29491200000000000"],["68719476736","8246337208320","255121057382400","3779571220480000","33218887680000000","191340793036800000","759605231616000000","2126008811520000000","4204265472000000000","5767168000000000000","5233704960000000000","2831155200000000000","692224000000000000"],["549755813888","71468255805440","2412053633433600","39307540692992000","383862702080000000","2487430309478400000","11285563441152000000","36850819399680000000","87448721817600000000","149946368000000000000","181435105280000000000","147220070400000000000","71991296000000000000","16056320000000000000"],["4398046511104","615726511554560","22512500578713600","400222232510464000","4299262263296000000","30954688295731200000","157997888176128000000","589613110394880000000","1632376140595200000000","3358798643200000000000","5080182947840000000000","5496215961600000000000","4031512576000000000000","1798307840000000000000","368640000000000000000"]],"degrees":[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14],"row_sums":["1","88","4944","230272","9646336","377223168","14053494784","505021038592","17648750493696","603187876200448","20244142278836224","669235437436403712","21842734363187347456","705147348962328444928","22549322413222882443264"],"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":[["4","-10","0"],["0","30","0"],["0","0","25"]]},"terms":[{"coeff":{"coeffs":[["32","240","600","500"],["0","240","1000","1000"],["0","0","200","500"]]},"deriv_order":0,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"}}],"properties":[{"id":192026,"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-9e43149565a3ac9a"]},{"id":192027,"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-9e43149565a3ac9a"]},{"id":192028,"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-9e43149565a3ac9a"]},{"id":192029,"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-9e43149565a3ac9a"]}],"region_exact":{},"incoming_edges":[{"id":"e-9e43149565a3ac9a","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-d99f7b722c4c442c","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":"154b4a648cdadaa37d409c15e9a8dbb441bb4d92dc2cf08a923d04228fb11257","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-815b996b7a3cb30d","role":"sequence","name":"basis_map_A013611 of Triangle of coefficients in expansion of (3+5x)^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":[{"id":"l-4d44e65ac2c484ed","kind":"fingerprint_related","node_a_id":"n-d99f7b722c4c442c","node_b_id":"n-fbc65803a7c7ccb7","fingerprint":"f6-23cc3a53c9ba374d","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-23cc3a53c9ba374d"},"other_id":"n-fbc65803a7c7ccb7","other":{"id":"n-fbc65803a7c7ccb7","name":"theta_plus_one_squared of basis_map_A013610 of Triangle of coefficients in expansion of (1+5x)^n.","first_index":0,"fingerprint":"f6-23cc3a53c9ba374d","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":0,"preview_rows":[["36","720","2025"],["216","6480","36450","54000"],["1296","51840","437400","1296000","1265625"]],"card_rows":[["36","720","2025"],["216","6480","36450","54000"],["1296","51840","437400","1296000","1265625"],["7776","388800","4374000","19440000","37968750","27337500"]],"row_sums_preview":["2781","97146","3052161","89516826","2501220141","67412171106","1766867858721","45290172451986"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-bbfc5a54ff13a785","steps":["basis A013610","(\u03b8+1)\u00b2"],"edges":["e-46de8f4df4fd1cb3","e-e6dead023820957c"],"text":"A013612 \u2192 basis A013610 \u2192 (\u03b8+1)\u00b2"},"headline":"A013612 \u2192 basis A013610 \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,          80;\n  2:        64,        1280,         3600;\n  3:       512,       15360,        86400,        128000;\n  4:      4096,      163840,      1382400,       4096000,        4000000;\n  5:     32768,     1638400,     18432000,      81920000,      160000000,      115200000;\n  6:    262144,    15728640,    221184000,    1310720000,     3840000000,     5529600000,      3136000000;\n  7:   2097152,   146800640,   2477260800,   18350080000,    71680000000,   154828800000,    175616000000,     81920000000;\n  8:  16777216,  1342177280,  26424115200,  234881024000,  1146880000000,  3303014400000,   5619712000000,   5242880000000,   2073600000000;\n  9: 134217728, 12079595520, 271790899200, 2818572288000, 16515072000000, 59454259200000, 134873088000000, 188743680000000, 149299200000000, 51200000000000;\n","recurrence_sentence":"The row polynomials satisfy (4+(30*n-10)*x+25*n^2*x^2)*P(n,x) = (32+(240*n+240)*x+(200*n^2+1000*n+600)*x^2+(500*n^2+1000*n+500)*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[((32 + (240*n + 240)*x + (200*n^2 + 1000*n + 600)*x^2 + (500*n^2 + 1000*n + 500)*x^3)*P[n - 1, x])/(4 + (30*n - 10)*x + 25*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] = ((32 + (240*n + 240)*x + (200*n^2 + 1000*n + 600)*x^2 + (500*n^2 + 1000*n + 500)*x^3)*v[i - 1])/(4 + (30*n - 10)*x + 25*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-c44f7114310405ad","steps":["basis A013611","(\u03b8+1)\u00b2"],"edges":["e-96f2019a633e1196","e-9e43149565a3ac9a"],"text":"A013622 \u2192 basis A013611 \u2192 (\u03b8+1)\u00b2"},"headline":"A013622 \u2192 basis A013611 \u2192 (\u03b8+1)\u00b2","chain":null}