{"id":"n-d55989cf30fde6ff","revision":1,"first_index":0,"fingerprint":"f6-a7224e2d1d90f526","name":"theta_plus_one of Triangle read by rows. T(n, k) = binomial(n, k)*binomial(n - 1, n - k - 1).","polytool_family":null,"interpretation":null,"rows_observed":15,"degree_pattern":"irregular","degree_slope":null,"degree_intercept":null,"palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":false,"rows":[["1"],["1"],["1","4"],["1","12","9"],["1","24","54","16"],["1","40","180","160","25"],["1","60","450","800","375","36"],["1","84","945","2800","2625","756","49"],["1","112","1764","7840","12250","7056","1372","64"],["1","144","3024","18816","44100","42336","16464","2304","81"],["1","180","4860","40320","132300","190512","123480","34560","3645","100"],["1","220","7425","79200","346500","698544","679140","316800","66825","5500","121"],["1","264","10890","145200","816750","2195424","2988216","2090880","735075","121000","7986","144"],["1","312","15444","251680","1769625","6115824","11099088","10872576","5733585","1573000","207636","11232","169"],["1","364","21294","416416","3578575","15459444","36072036","47114496","34783749","14314300","3149146","340704","15379","196"]],"degrees":[0,0,1,2,3,4,5,6,7,8,9,10,11,12,13],"row_sums":["1","1","5","22","95","406","1722","7260","30459","127270","529958","2200276","9111830","37650172","155266100"],"nonn":true,"definitions":[{"position":0,"source":"derived:theta_plus_one","source_kind":"derived","equality_edge_id":null,"recurrence":{"first_index":0,"initial_polynomials":[["1"],["1"]],"recurrence":{"denominator":{"coeffs":[["0","0"],["1","0"],["-1","1"],["0","-2"],["0","1"]]},"terms":[{"coeff":{"coeffs":[["0","-4","-2"],["1","10","5"],["-1","-5","-2"],["0","-2","-4"],["0","1","3"]]},"deriv_order":0,"offset":1,"sign":"none"},{"coeff":{"coeffs":[["0","1","0","-1"],["0","-2","0","2"],["0","0","0","0"],["0","0","2","-2"]]},"deriv_order":1,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"}}],"properties":[{"id":164002,"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-65f24507891d8ee5"]},{"id":164003,"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-65f24507891d8ee5"]},{"id":164004,"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-65f24507891d8ee5"]},{"id":164005,"position":3,"claim":"interlacing","from_index":2,"region_key":null,"provenance":"edges","claim_data":{"claim":"interlacing","from_index":2,"kind":"weak","orientation":"unordered"},"label":"Interlacing of consecutive rows","description":"Rows P_(m-1), P_m interlace (weak, unordered) for every m >= 2","region_exact":null,"edges":["e-65f24507891d8ee5"]}],"region_exact":{},"incoming_edges":[{"id":"e-65f24507891d8ee5","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-d55989cf30fde6ff","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":"873937c767773d8d82009f7a9a728e044632151190295e850c0146fe5e9ea5e4","verification_polytool":"replayed","verification_lean":"not_attempted","claims":[{"claim":"real_rooted","from_index":0,"region":{"upper":{"closed":true,"value":"0"}}},{"claim":"interlacing","from_index":2,"kind":"weak","orientation":"unordered"}],"premises":[{"position":0,"node_id":"n-b1b19325e0094118","role":"sequence","name":"Triangle read by rows. T(n, k) = binomial(n, k)*binomial(n - 1, n - k - 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":[{"id":"l-4b26fda7ab7feb3d","kind":"fingerprint_related","node_a_id":"n-ab83e9966290e807","node_b_id":"n-d55989cf30fde6ff","fingerprint":"f6-a7224e2d1d90f526","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-a7224e2d1d90f526"},"other_id":"n-ab83e9966290e807","other":{"id":"n-ab83e9966290e807","name":"theta_plus_one_squared of narayana","first_index":1,"fingerprint":"f6-a7224e2d1d90f526","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":0,"preview_rows":[["1"],["1","4"],["1","12","9"]],"card_rows":[["1"],["1","4"],["1","12","9"],["1","24","54","16"]],"row_sums_preview":["1","5","22","95","406","1722","7260","30459"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-2a05d8340c329b21","steps":["(\u03b8+1)\u00b2"],"edges":["e-1770020eb0a9c55e"],"text":"narayana \u2192 (\u03b8+1)\u00b2"},"headline":"narayana \u2192 (\u03b8+1)\u00b2","chain":null}},{"id":"l-ce738d8c090ba79c","kind":"fingerprint_related","node_a_id":"n-4401d8c28f92f0bd","node_b_id":"n-d55989cf30fde6ff","fingerprint":"f6-a7224e2d1d90f526","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-a7224e2d1d90f526"},"other_id":"n-4401d8c28f92f0bd","other":{"id":"n-4401d8c28f92f0bd","name":"theta_plus_one_squared of Triangle of Narayana numbers T(n,k) = C(n-1,k-1)*C(n,k-1)/k with 1 <= k <= n, read by rows. Also called the Catalan triangle.","first_index":0,"fingerprint":"f6-a7224e2d1d90f526","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":0,"preview_rows":[["1"],["1","4"],["1","12","9"]],"card_rows":[["1"],["1","4"],["1","12","9"],["1","24","54","16"]],"row_sums_preview":["1","5","22","95","406","1722","7260","30459"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-c2eaf5212a4dec13","steps":["(\u03b8+1)\u00b2"],"edges":["e-f46dab037b613258"],"text":"A001263 \u2192 (\u03b8+1)\u00b2"},"headline":"A001263 \u2192 (\u03b8+1)\u00b2","chain":null}},{"id":"l-a4a1cf3ee51c46d8","kind":"oeis_match","node_a_id":"n-d55989cf30fde6ff","node_b_id":null,"fingerprint":null,"oeis_id":"A141222","in_catalog":0,"target":"row_sums","row_alignment":null,"rows_matched":null,"oeis_nonn":1,"details":{"alignment":{"first_row":1,"low_zeros_dropped":false,"oeis_offset":0,"oeis_term_position":0,"target":"row_sums","terms_compared":14},"in_catalog":false,"oeis_id":"A141222","oeis_nonn":true},"other_id":null,"other":null}],"oeis":[{"oeis_id":"A141222","relation":"match_row_sums"}],"bfiles":[{"node_id":"n-d55989cf30fde6ff","oeis_id":"A141222","target":"row_sums","oeis_offset":0,"first_row":1,"rows_used":200,"terms_matched":14,"oeis_terms_agreeing":26,"oeis_terms_available":26,"term_count":200,"raw_size":13112}],"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: 1;\n  2: 1,   4;\n  3: 1,  12,    9;\n  4: 1,  24,   54,    16;\n  5: 1,  40,  180,   160,    25;\n  6: 1,  60,  450,   800,   375,    36;\n  7: 1,  84,  945,  2800,  2625,   756,    49;\n  8: 1, 112, 1764,  7840, 12250,  7056,  1372,   64;\n  9: 1, 144, 3024, 18816, 44100, 42336, 16464, 2304, 81;\n","recurrence_sentence":"The row polynomials satisfy -(n^2-n-(n^4-2*n^3+n^2)*x)*P(n,x) = -(n^2-n-(n^4-2*n^3-5*n^2+10*n-4)*x-(3*n^4-4*n^3-2*n^2+5*n-2)*x^2)*P(n-1,x) - ((2*n-1)*x-2*n^3*x^2+(2*n^3-2*n+1)*x^3)*d/dx P(n-1,x) for n >= 2, with P(0,x) = 1, P(1,x) = 1.","mathematica":"P[0, x_] := 1;\nP[1, x_] := 1;\nP[n_, x_] := P[n, x] = Expand[Together[(-(n^2 - n - (n^4 - 2*n^3 - 5*n^2 + 10*n - 4)*x - (3*n^4 - 4*n^3 - 2*n^2 + 5*n - 2)*x^2)*P[n - 1, x] - ((2*n - 1)*x - 2*n^3*x^2 + (2*n^3 - 2*n + 1)*x^3)*D[P[n - 1, x], x])/(-n^2 + n + (n^4 - 2*n^3 + n^2)*x)]];\nTable[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten","pari":"P(N) = {my(v = vector(N + 1)); v[1] = 1; v[2] = 1; for(n = 2, N, my(i = n + 1); v[i] = (-(n^2 - n - (n^4 - 2*n^3 - 5*n^2 + 10*n - 4)*x - (3*n^4 - 4*n^3 - 2*n^2 + 5*n - 2)*x^2)*v[i - 1] - ((2*n - 1)*x - 2*n^3*x^2 + (2*n^3 - 2*n + 1)*x^3)*deriv(v[i - 1], x))/(-n^2 + n + (n^4 - 2*n^3 + n^2)*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-b1b19325e0094118","steps":["\u03b8+1"],"edges":["e-65f24507891d8ee5"],"text":"A367270 \u2192 \u03b8+1"},"headline":"A367270 \u2192 \u03b8+1","chain":null}