{"id":"n-a1b1c992dac01d83","revision":0,"first_index":0,"fingerprint":"f6-572def99d14f8fb6","name":"theta_plus_one of Triangle whose (i,j)-th entry is binomial(i,j)*10^(i-j)*12^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":true,"rows":[["100","480","432"],["1000","7200","12960","6912"],["10000","96000","259200","276480","103680"],["100000","1200000","4320000","6912000","5184000","1492992"],["1000000","14400000","64800000","138240000","155520000","89579520","20901888"],["10000000","168000000","907200000","2419200000","3628800000","3135283200","1463132160","286654464"],["100000000","1920000000","12096000000","38707200000","72576000000","83607552000","58525286400","22932357120","3869835264"],["1000000000","21600000000","155520000000","580608000000","1306368000000","1881169920000","1755758592000","1031956070400","348285173760","51597803520"],["10000000000","240000000000","1944000000000","8294400000000","21772800000000","37623398400000","43893964800000","34398535680000","17414258688000","5159780352000","681091006464"],["100000000000","2640000000000","23760000000000","114048000000000","342144000000000","689762304000000","965667225600000","945959731200000","638522818560000","283787919360000","74920010711040","8916100448256"],["1000000000000","28800000000000","285120000000000","1520640000000000","5132160000000000","11824496640000000","19313344512000000","22703033548800000","19155684556800000","11351516774400000","4495200642662400","1069932053790720","115909305827328"],["10000000000000","312000000000000","3369600000000000","19768320000000000","74131200000000000","192148070400000000","358676398080000000","491899060224000000","498047798476800000","368924295168000000","194792027848704000","69545583496396800","15068209757552640","1497904875307008"],["100000000000000","3360000000000000","39312000000000000","251596800000000000","1037836800000000000","2988969984000000000","6276836966400000000","9837981204480000000","11621115297792000000","10329880264704000000","6817720974704640000","3245460563165184000","1054774683028684800","209706682542981120","19258776968232960"],["1000000000000000","36000000000000000","453600000000000000","3144960000000000000","14152320000000000000","44834549760000000000","104613949440000000000","184462147584000000000","249023899238400000000","258247006617600000000","204531629241139200000","121704771118694400000","52738734151434240000","15728001190723584000","2888816545234944000","246512345193381888"],["10000000000000000","384000000000000000","5184000000000000000","38707200000000000000","188697600000000000000","652138905600000000000","1673823191040000000000","3279327068160000000000","4980477984768000000000","5902788722688000000000","5454176779763712000000","3894552675798220800000","2109549366057369600000","838826730171924480000","231105323618795520000","39441975230941102080","3143032401215619072"]],"degrees":[2,3,4,5,6,7,8,9,10,11,12,13,14,15,16],"row_sums":["1012","28072","745360","19208992","484441408","12018269824","294334230784","7133863559680","171432228926464","4090228109879296","96996838034280448","2288190468326760448","53733910997785722880","1256807897232419749888","29292334555298179121152"],"nonn":true,"definitions":[{"position":0,"source":"derived:theta_plus_one","source_kind":"derived","equality_edge_id":null,"recurrence":{"first_index":0,"initial_polynomials":[["100","480","432"]],"recurrence":{"denominator":{"coeffs":[["5","12"],["0","6"]]},"terms":[{"coeff":{"coeffs":[["50","240","216"],["0","60","72"]]},"deriv_order":0,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"}}],"properties":[{"id":193428,"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-cab86ae5c93bac18"]},{"id":193429,"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-cab86ae5c93bac18"]},{"id":193430,"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-cab86ae5c93bac18"]},{"id":193431,"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-cab86ae5c93bac18"]}],"region_exact":{},"incoming_edges":[{"id":"e-cab86ae5c93bac18","operation":{"index_shift":0,"operation":"preserver_image","operator":"{\"shape\":\"euler_polynomial\",\"coefficients\":[{\"numerator\":[\"1\"],\"denominator\":[\"1\"]},{\"numerator\":[\"1\"],\"denominator\":[\"1\"]}]}"},"operation_detail":"euler_polynomial, shift 0","conclusion_id":"n-a1b1c992dac01d83","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":"cd3593b9fc72b1be8bd1ce6d9444d6dc32506369fac78c33ffd2cab4dadbe4a8","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-29d7c5a1e19b9d74","role":"sequence","name":"Triangle whose (i,j)-th entry is binomial(i,j)*10^(i-j)*12^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-a3c84fde79fd9ead","kind":"fingerprint_related","node_a_id":"n-b8540c42f24ca890","node_b_id":"n-a1b1c992dac01d83","fingerprint":"f6-572def99d14f8fb6","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-572def99d14f8fb6"},"other_id":"n-b8540c42f24ca890","other":{"id":"n-b8540c42f24ca890","name":"theta_plus_one of basis_map_A013609 of Triangle of coefficients in expansion of (2+3x)^n.","first_index":0,"fingerprint":"f6-572def99d14f8fb6","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["1"],["5","12"],["25","120","108"]],"card_rows":[["1"],["5","12"],["25","120","108"],["125","900","1620","864"]],"row_sums_preview":["1","17","253","3509","46585","600281","7569397","93892733"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-01cea33b6485d9cc","steps":["basis A013610","\u03b8+1"],"edges":["e-32b2129ae50cd22c","e-186346ec1b78b1bf"],"text":"A038220 \u2192 basis A013610 \u2192 \u03b8+1"},"headline":"A038220 \u2192 basis A013610 \u2192 \u03b8+1","chain":null}},{"id":"l-e4fe69b329dc84cb","kind":"fingerprint_related","node_a_id":"n-a1b1c992dac01d83","node_b_id":"n-1d55f39abb48a994","fingerprint":"f6-572def99d14f8fb6","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-572def99d14f8fb6"},"other_id":"n-1d55f39abb48a994","other":{"id":"n-1d55f39abb48a994","name":"theta_plus_one of basis_map_A013610 of Triangle whose (i,j)-th entry is binomial(i,j)*12^(i-j)*8^j.","first_index":0,"fingerprint":"f6-572def99d14f8fb6","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["400","1920","1728"],["8000","57600","103680","55296"],["160000","1536000","4147200","4423680","1658880"]],"card_rows":[["400","1920","1728"],["8000","57600","103680","55296"],["160000","1536000","4147200","4423680","1658880"],["3200000","38400000","138240000","221184000","165888000","47775744"]],"row_sums_preview":["4048","224576","11925760","614687744","31004250112","1538338537472","75349563080704","3652538142556160"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-6874f381e9c7e0a2","steps":["basis A013610","\u03b8+1"],"edges":["e-889e30cc3b49f76a","e-28b8093277bae08a"],"text":"A038334 \u2192 basis A013610 \u2192 \u03b8+1"},"headline":"A038334 \u2192 basis A013610 \u2192 \u03b8+1","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:          100,           480,            432;\n  1:         1000,          7200,          12960,            6912;\n  2:        10000,         96000,         259200,          276480,          103680;\n  3:       100000,       1200000,        4320000,         6912000,         5184000,         1492992;\n  4:      1000000,      14400000,       64800000,       138240000,       155520000,        89579520,        20901888;\n  5:     10000000,     168000000,      907200000,      2419200000,      3628800000,      3135283200,      1463132160,       286654464;\n  6:    100000000,    1920000000,    12096000000,     38707200000,     72576000000,     83607552000,     58525286400,     22932357120,      3869835264;\n  7:   1000000000,   21600000000,   155520000000,    580608000000,   1306368000000,   1881169920000,   1755758592000,   1031956070400,    348285173760,     51597803520;\n  8:  10000000000,  240000000000,  1944000000000,   8294400000000,  21772800000000,  37623398400000,  43893964800000,  34398535680000,  17414258688000,   5159780352000,   681091006464;\n  9: 100000000000, 2640000000000, 23760000000000, 114048000000000, 342144000000000, 689762304000000, 965667225600000, 945959731200000, 638522818560000, 283787919360000, 74920010711040, 8916100448256;\n","recurrence_sentence":"The row polynomials satisfy (5+(6*n+12)*x)*P(n,x) = (50+(60*n+240)*x+(72*n+216)*x^2)*P(n-1,x) for n >= 1, with P(0,x) = 100 + 480*x + 432*x^2.","mathematica":"P[0, x_] := 100 + 480*x + 432*x^2;\nP[n_, x_] := P[n, x] = Expand[Together[((50 + (60*n + 240)*x + (72*n + 216)*x^2)*P[n - 1, x])/(5 + (6*n + 12)*x)]];\nTable[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten","pari":"P(N) = {my(v = vector(N + 1)); v[1] = 100 + 480*x + 432*x^2; for(n = 1, N, my(i = n + 1); v[i] = ((50 + (60*n + 240)*x + (72*n + 216)*x^2)*v[i - 1])/(5 + (6*n + 12)*x)); 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-29d7c5a1e19b9d74","steps":["\u03b8+1"],"edges":["e-cab86ae5c93bac18"],"text":"A038314 \u2192 \u03b8+1"},"headline":"A038314 \u2192 \u03b8+1","chain":null}