{"id":"n-d04d98855e40597e","revision":0,"first_index":0,"fingerprint":"f6-18b5ed15f2b69ef1","name":"theta_plus_one_squared of basis_map_A013611 of Triangle whose (i,j)-th entry is binomial(i,j)*12^(i-j)*2^j.","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":[["196","896","576"],["2744","18816","24192","8192"],["38416","351232","677376","458752","102400"],["537824","6146560","15805440","16056320","7168000","1179648"],["7529536","103262208","331914240","449576960","301056000","99090432","12845056"],["105413504","1686616064","6505519104","11014635520","9834496000","4855431168","1258815488","134217728"],["1475789056","26985857024","121436356608","246727835648","275365888000","181269430272","70493667328","15032385536","1358954496"],["20661046784","425027248128","2185854418944","5181284548608","6939220377600","5709987053568","2960734027776","947040288768","171228266496","13421772800"],["289254654976","6611534970880","38252452331520","103625690972160","161915142144000","159879637499904","103625690972160","44195213475840","11985978654720","1879048192000","129922760704"],["4049565169664","101817638551552","654541962117120","1994794551214080","3562133127168000","4103577362497536","3191671281942528","1701515718819840","615280237608960","144686710784000","20008105148416","1236950581248"],["56693912375296","1555033025150976","10996304963567616","37236164955996160","74804795670528000","98485856699940864","89366795894390784","57170928152346624","25841769979576320","8102455803904000","1680680832466944","207807697649664","11613591568384"],["793714773254144","23584667548123136","181938863942664192","677698202199130112","1512719201337344000","2240553239923654656","2323536693254160384","1734184820621180928","940640427256578048","368661739077632000","101961303836327936","18910500486119424","2113673665445888","107752139522048"],["11112006825558016","355584218417856512","2971668111063515136","12075349784639045632","29649296346211942400","48794270558337368064","56926648984726929408","48557174977393065984","30727587290381549568","14451540171843174400","4996103887980068864","1235486031759802368","207140019213697024","21119419346321408","989560464998400"],["155568095557812224","5333763276267847680","48003869486410629120","211318621231183298560","566032021154955264000","1024679681725084729344","1328288476310295019520","1274625843156567982080","921827618711446487040","505803906014511104000","209836363295162892288","64863016667389624320","14499801344958791680","2217539031363747840","207807697649664000","9007199254740992"],["2177953337809371136","79650864925599858688","768061911782570065920","3641182396598850682880","10565931061559164928000","20866204427856270852096","29753661869350608437248","31724020985230136442880","25811173323920501637120","16185724992464355328000","7833890896352747978752","2905863146699055169536","811988875317692334080","165576247675159838720","23274462136762368000","2017612633061982208","81346268269379584"]],"degrees":[2,3,4,5,6,7,8,9,10,11,12,13,14,15,16],"row_sums":["1668","53944","1628176","46893792","1305274432","35395144576","940146163968","24554459049472","632389566628864","16095313211602944","405516901179461632","10127404800061136896","250981071368604893184","6177702904398059634688","151140482374108616654848"],"nonn":true,"definitions":[{"position":0,"source":"derived:theta_plus_one_squared","source_kind":"derived","equality_edge_id":null,"recurrence":{"first_index":0,"initial_polynomials":[["196","896","576"]],"recurrence":{"denominator":{"coeffs":[["49","140","64"],["0","84","64"],["0","0","16"]]},"terms":[{"coeff":{"coeffs":[["686","3528","3808","1152"],["0","1176","2016","768"],["0","0","224","128"]]},"deriv_order":0,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"}}],"properties":[{"id":193652,"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-51f2c66c91c88e90"]},{"id":193653,"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-51f2c66c91c88e90"]},{"id":193654,"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-51f2c66c91c88e90"]},{"id":193655,"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-51f2c66c91c88e90"]}],"region_exact":{},"incoming_edges":[{"id":"e-51f2c66c91c88e90","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-d04d98855e40597e","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":"a093a085a61012a1b2d7f58c402faa13d40947d03dae8e4774736488b3b20d4e","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-8afe6c02ab69bf6d","role":"sequence","name":"basis_map_A013611 of Triangle whose (i,j)-th entry is binomial(i,j)*12^(i-j)*2^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-1164c486310e5d02","kind":"fingerprint_related","node_a_id":"n-6266b8f8b17b926c","node_b_id":"n-d04d98855e40597e","fingerprint":"f6-18b5ed15f2b69ef1","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-18b5ed15f2b69ef1"},"other_id":"n-6266b8f8b17b926c","other":{"id":"n-6266b8f8b17b926c","name":"theta_plus_one_squared of basis_map_A013611 of Triangle whose (i,j)-th entry is binomial(i,j)*6^(i-j).","first_index":0,"fingerprint":"f6-18b5ed15f2b69ef1","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":0,"preview_rows":[["1"],["7","16"],["49","224","144"]],"card_rows":[["1"],["7","16"],["49","224","144"],["343","2352","3024","1024"]],"row_sums_preview":["1","23","417","6743","101761","1465431","20394913","276524567"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-22693abd431a71f2","steps":["basis A013611","(\u03b8+1)\u00b2"],"edges":["e-1b27bff9bafb5517","e-038f08ab2c78f1d5"],"text":"A038255 \u2192 basis A013611 \u2192 (\u03b8+1)\u00b2"},"headline":"A038255 \u2192 basis A013611 \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:           196,             896,             576;\n  1:          2744,           18816,           24192,             8192;\n  2:         38416,          351232,          677376,           458752,           102400;\n  3:        537824,         6146560,        15805440,         16056320,          7168000,          1179648;\n  4:       7529536,       103262208,       331914240,        449576960,        301056000,         99090432,         12845056;\n  5:     105413504,      1686616064,      6505519104,      11014635520,       9834496000,       4855431168,       1258815488,        134217728;\n  6:    1475789056,     26985857024,    121436356608,     246727835648,     275365888000,     181269430272,      70493667328,      15032385536,      1358954496;\n  7:   20661046784,    425027248128,   2185854418944,    5181284548608,    6939220377600,    5709987053568,    2960734027776,     947040288768,    171228266496,     13421772800;\n  8:  289254654976,   6611534970880,  38252452331520,  103625690972160,  161915142144000,  159879637499904,  103625690972160,   44195213475840,  11985978654720,   1879048192000,   129922760704;\n  9: 4049565169664, 101817638551552, 654541962117120, 1994794551214080, 3562133127168000, 4103577362497536, 3191671281942528, 1701515718819840, 615280237608960, 144686710784000, 20008105148416, 1236950581248;\n","recurrence_sentence":"The row polynomials satisfy (49+(84*n+140)*x+(16*n^2+64*n+64)*x^2)*P(n,x) = (686+(1176*n+3528)*x+(224*n^2+2016*n+3808)*x^2+(128*n^2+768*n+1152)*x^3)*P(n-1,x) for n >= 1, with P(0,x) = 196 + 896*x + 576*x^2.","mathematica":"P[0, x_] := 196 + 896*x + 576*x^2;\nP[n_, x_] := P[n, x] = Expand[Together[((686 + (1176*n + 3528)*x + (224*n^2 + 2016*n + 3808)*x^2 + (128*n^2 + 768*n + 1152)*x^3)*P[n - 1, x])/(49 + (84*n + 140)*x + (16*n^2 + 64*n + 64)*x^2)]];\nTable[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten","pari":"P(N) = {my(v = vector(N + 1)); v[1] = 196 + 896*x + 576*x^2; for(n = 1, N, my(i = n + 1); v[i] = ((686 + (1176*n + 3528)*x + (224*n^2 + 2016*n + 3808)*x^2 + (128*n^2 + 768*n + 1152)*x^3)*v[i - 1])/(49 + (84*n + 140)*x + (16*n^2 + 64*n + 64)*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-ea9b82d727693fb6","steps":["basis A013611","(\u03b8+1)\u00b2"],"edges":["e-0eb3975e7d8d74fc","e-51f2c66c91c88e90"],"text":"A038328 \u2192 basis A013611 \u2192 (\u03b8+1)\u00b2"},"headline":"A038328 \u2192 basis A013611 \u2192 (\u03b8+1)\u00b2","chain":null}
