{"id":"n-30f01afcd7e05a42","revision":0,"first_index":0,"fingerprint":"f6-51a5affbafca912b","name":"theta_plus_one_squared of basis_map_A013611 of Triangle whose (i,j)-th entry is binomial(i,j)*11^(i-j)*5^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":[["256","2560","3600"],["4096","61440","172800","128000"],["65536","1310720","5529600","8192000","4000000"],["1048576","26214400","147456000","327680000","320000000","115200000"],["16777216","503316480","3538944000","10485760000","15360000000","11059200000","3136000000"],["268435456","9395240960","79272345600","293601280000","573440000000","619315200000","351232000000","81920000000"],["4294967296","171798691840","1691143372800","7516192768000","18350080000000","26424115200000","22478848000000","10485760000000","2073600000000"],["68719476736","3092376453120","34789235097600","180388626432000","528482304000000","951268147200000","1078984704000000","754974720000000","298598400000000","51200000000000"],["1099511627776","54975581388800","695784701952000","4123168604160000","14092861440000000","30440580710400000","43159388160000000","40265318400000000","23887872000000000","8192000000000000","1239040000000000"],["17592186044416","967570232442880","13606456393728000","90709709291520000","354334801920000000","892923700838400000","1519210463232000000","1771674009600000000","1401421824000000000","720896000000000000","218071040000000000","29491200000000000"],["281474976710656","16888498602639360","261243962759577600","1935140464885760000","8504035246080000000","24491621508710400000","48614734823424000000","68032281968640000000","67268247552000000000","46137344000000000000","20934819840000000000","5662310400000000000","692224000000000000"],["4503599627370496","292733975779082240","4939885841272012800","40250921669623808000","196537703464960000000","636782159226470400000","1444552120467456000000","2358452441579520000000","2798359098163200000000","2399141888000000000000","1451480842240000000000","588880281600000000000","143982592000000000000","16056320000000000000"],["72057594037927936","5044031582654955520","92211202370410905600","819655132181430272000","4402444557615104000000","15848800407414374400000","40447459373088768000000","75470478130544640000000","104472072998092800000000","107481556582400000000000","81282927165440000000000","43969727692800000000000","16126050304000000000000","3596615680000000000000","368640000000000000000"],["1152921504606846976","86469112845513523200","1702360659146047488000","16393102643628605440000","96053335802511360000000","380371209777944985600000","1078598916615700480000000","2264114343916339200000000","3581899645648896000000000","4299262263296000000000000","3901580503941120000000000","2638183661568000000000000","1290084024320000000000000","431593881600000000000000","88473600000000000000000","8388608000000000000000"],["18446744073709551616","1475739525896764129280","31128880624384868352000","322818021289917153280000","2049137830453575680000000","8852275427559446937600000","27612132265361932288000000","64401474671398092800000000","114620788660764672000000000","157230162771968000000000000","166467434834821120000000000","135075003472281600000000000","82565377556480000000000000","36829344563200000000000000","11324620800000000000000000","2147483648000000000000000","189399040000000000000000"]],"degrees":[2,3,4,5,6,7,8,9,10,11,12,13,14,15,16],"row_sums":["6416","366336","19097856","937598976","44099997696","2008444502016","89195832999936","3881847232659456","166152089109528576","7013324367694135296","292551173740079087616","12079713491827908673536","494383755315124220461056","20076787079823636798898176","809720076630472712200912896"],"nonn":true,"definitions":[{"position":0,"source":"derived:theta_plus_one_squared","source_kind":"derived","equality_edge_id":null,"recurrence":{"first_index":0,"initial_polynomials":[["256","2560","3600"]],"recurrence":{"denominator":{"coeffs":[["16","100","100"],["0","60","100"],["0","0","25"]]},"terms":[{"coeff":{"coeffs":[["256","2880","6800","4500"],["0","960","3600","3000"],["0","0","400","500"]]},"deriv_order":0,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"}}],"properties":[{"id":194715,"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-12404c4861573b72"]},{"id":194716,"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-12404c4861573b72"]},{"id":194717,"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-12404c4861573b72"]},{"id":194718,"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-12404c4861573b72"]}],"region_exact":{},"incoming_edges":[{"id":"e-12404c4861573b72","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-30f01afcd7e05a42","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":"84bd396e6457a105dd5fcc21bc87dc05c05fd9af3f0d4d9bf65165d7afada154","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-93016690ff7736f4","role":"sequence","name":"basis_map_A013611 of Triangle whose (i,j)-th entry is binomial(i,j)*11^(i-j)*5^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-cb02fa24a4e709ec","kind":"fingerprint_related","node_a_id":"n-2de0c804fb24112d","node_b_id":"n-30f01afcd7e05a42","fingerprint":"f6-51a5affbafca912b","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-51a5affbafca912b"},"other_id":"n-2de0c804fb24112d","other":{"id":"n-2de0c804fb24112d","name":"theta_plus_one_squared of basis_map_A013609 of Triangle of coefficients in expansion of (3+5x)^n.","first_index":0,"fingerprint":"f6-51a5affbafca912b","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":0,"preview_rows":[["1"],["8","40"],["64","640","900"]],"card_rows":[["1"],["8","40"],["64","640","900"],["512","7680","21600","16000"]],"row_sums_preview":["1","48","1604","45792","1193616","29299968","689062464","15690972672"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-c44f7114310405ad","steps":["basis A013609","(\u03b8+1)\u00b2"],"edges":["e-c5caeaebd0a50686","e-38961f075b6ef36d"],"text":"A013622 \u2192 basis A013609 \u2192 (\u03b8+1)\u00b2"},"headline":"A013622 \u2192 basis A013609 \u2192 (\u03b8+1)\u00b2","chain":null}},{"id":"l-f7829d4c0e24cec9","kind":"fingerprint_related","node_a_id":"n-5a6fba4932c0e69a","node_b_id":"n-30f01afcd7e05a42","fingerprint":"f6-51a5affbafca912b","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-51a5affbafca912b"},"other_id":"n-5a6fba4932c0e69a","other":{"id":"n-5a6fba4932c0e69a","name":"theta_plus_one_squared of basis_map_A013610 of Triangle whose (i,j)-th entry is binomial(i,j)*7^(i-j)*5^j.","first_index":0,"fingerprint":"f6-51a5affbafca912b","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":0,"preview_rows":[["1"],["12","60"],["144","1440","2025"]],"card_rows":[["1"],["12","60"],["144","1440","2025"],["1728","25920","72900","54000"]],"row_sums_preview":["1","72","3609","154548","6042681","222496632","7848852129","268094978388"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-a6a633b989da1879","steps":["basis A013610","(\u03b8+1)\u00b2"],"edges":["e-cc1debec4a411a9c","e-d839bc0dffa677f5"],"text":"A038271 \u2192 basis A013610 \u2192 (\u03b8+1)\u00b2"},"headline":"A038271 \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:            256,            2560,              3600;\n  1:           4096,           61440,            172800,            128000;\n  2:          65536,         1310720,           5529600,           8192000,            4000000;\n  3:        1048576,        26214400,         147456000,         327680000,          320000000,          115200000;\n  4:       16777216,       503316480,        3538944000,       10485760000,        15360000000,        11059200000,          3136000000;\n  5:      268435456,      9395240960,       79272345600,      293601280000,       573440000000,       619315200000,        351232000000,         81920000000;\n  6:     4294967296,    171798691840,     1691143372800,     7516192768000,     18350080000000,     26424115200000,      22478848000000,      10485760000000,       2073600000000;\n  7:    68719476736,   3092376453120,    34789235097600,   180388626432000,    528482304000000,    951268147200000,    1078984704000000,     754974720000000,     298598400000000,     51200000000000;\n  8:  1099511627776,  54975581388800,   695784701952000,  4123168604160000,  14092861440000000,  30440580710400000,   43159388160000000,   40265318400000000,   23887872000000000,   8192000000000000,   1239040000000000;\n  9: 17592186044416, 967570232442880, 13606456393728000, 90709709291520000, 354334801920000000, 892923700838400000, 1519210463232000000, 1771674009600000000, 1401421824000000000, 720896000000000000, 218071040000000000, 29491200000000000;\n","recurrence_sentence":"The row polynomials satisfy (16+(60*n+100)*x+(25*n^2+100*n+100)*x^2)*P(n,x) = (256+(960*n+2880)*x+(400*n^2+3600*n+6800)*x^2+(500*n^2+3000*n+4500)*x^3)*P(n-1,x) for n >= 1, with P(0,x) = 256 + 2560*x + 3600*x^2.","mathematica":"P[0, x_] := 256 + 2560*x + 3600*x^2;\nP[n_, x_] := P[n, x] = Expand[Together[((256 + (960*n + 2880)*x + (400*n^2 + 3600*n + 6800)*x^2 + (500*n^2 + 3000*n + 4500)*x^3)*P[n - 1, x])/(16 + (60*n + 100)*x + (25*n^2 + 100*n + 100)*x^2)]];\nTable[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten","pari":"P(N) = {my(v = vector(N + 1)); v[1] = 256 + 2560*x + 3600*x^2; for(n = 1, N, my(i = n + 1); v[i] = ((256 + (960*n + 2880)*x + (400*n^2 + 3600*n + 6800)*x^2 + (500*n^2 + 3000*n + 4500)*x^3)*v[i - 1])/(16 + (60*n + 100)*x + (25*n^2 + 100*n + 100)*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-2bc1923905556bf6","steps":["basis A013611","(\u03b8+1)\u00b2"],"edges":["e-fd91849598d32af1","e-12404c4861573b72"],"text":"A038319 \u2192 basis A013611 \u2192 (\u03b8+1)\u00b2"},"headline":"A038319 \u2192 basis A013611 \u2192 (\u03b8+1)\u00b2","chain":null}
