{"id":"n-02b8fac5a8097d07","revision":0,"first_index":0,"fingerprint":"f6-6443481be5042dac","name":"shift_x_plus_one of derivative of Triangle T(n,k), 0 <= k <= n, read by rows, giving coefficients of the polynomial (x+1)(x+2)...(x+n), expanded in increasing powers of x. T(n,k) is also the unsigned Stirling number |s(n+1, k+1)|, denoting the number of permutations on n+1 elements that contain exactly k+1 cycles.","polytool_family":null,"interpretation":null,"rows_observed":15,"degree_pattern":"affine","degree_slope":1,"degree_intercept":-1,"palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":false,"rows":[[],["1"],["5","2"],["26","18","3"],["154","142","42","4"],["1044","1160","465","80","5"],["8028","10208","4995","1180","135","6"],["69264","97720","55272","16100","2555","210","7"],["663696","1018008","644028","218596","43120","4956","308","8"],["6999840","11507472","7967292","3044664","704445","101304","8862","432","9"],["80627040","140581872","104901420","44114360","11554725","1959678","215670","14880","585","10"],["1007441280","1848236544","1469689848","669240880","193799650","37381806","4874331","425040","23760","770","11"],["13575738240","26041957632","21878322840","10659717904","3355946550","718523058","106978410","11096184","787050","36410","990","12"],["196287356160","391738883328","345359456208","178407147776","60307899080","14086458672","2335974641","277607616","23501907","1384240","53911","1248","13"],["3031488633600","6268657962240","5768000168112","3136586491584","1127627420920","283666358976","51473821399","6833799544","664837173","46876830","2331329","77532","1547","14"]],"degrees":[-1,0,1,2,3,4,5,6,7,8,9,10,11,12,13],"row_sums":["0","1","7","47","342","2754","24552","241128","2592720","30334320","383970240","5231113920","76349105280","1188825724800","19675048780800"],"nonn":true,"definitions":[{"position":0,"source":"derived:shift_x_plus_one","source_kind":"derived","equality_edge_id":null,"recurrence":{"first_index":0,"initial_polynomials":[["0"],["1"]],"recurrence":{"terms":[{"coeff":{"coeffs":[["0","0","-1"],["0","-2","0"],["-1","0","0"]]},"deriv_order":0,"offset":2,"sign":"none"},{"coeff":{"coeffs":[["1","2"],["2","0"]]},"deriv_order":0,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"}}],"properties":[{"id":179035,"position":0,"claim":"real_rooted","from_index":0,"region_key":"(-inf, -1]","provenance":"edges","claim_data":{"claim":"real_rooted","from_index":0,"region":{"upper":{"closed":true,"value":"-1"}}},"label":"Real-rooted","description":"Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, -1]","region_exact":null,"edges":["e-0c3459bae8cf1790"]},{"id":179036,"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-0c3459bae8cf1790","e-2b1911b9eb616677"]},{"id":179037,"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-0c3459bae8cf1790","e-2b1911b9eb616677"]},{"id":179038,"position":3,"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-2b1911b9eb616677"]},{"id":179039,"position":4,"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-2b1911b9eb616677"]}],"region_exact":{},"incoming_edges":[{"id":"e-0c3459bae8cf1790","operation":{"index_shift":0,"operation":"preserver_image","operator":"{\"shape\":\"affine\",\"slope\":\"1\",\"intercept\":\"1\"}"},"operation_detail":"affine, shift 0","conclusion_id":"n-02b8fac5a8097d07","premise_count":1,"theorem":"a real affine substitution x -> a x + b (a != 0), the reversal x^d p(1/x) (d >= deg p) and a real Mobius map (c x + e)^d p((a x + b)/(c x + e)) (a e - b c != 0, d >= deg p) map real roots to real roots: a root r of p gives the root (r - b)/a, 1/r (and 0) or (e r - b)/(a - c r) (and -e/c); for an input row that is a polynomial the degree condition holds automatically","lean_route":null,"lean_declarations":["Polynomial.Splits.comp_of_degree_le_one"],"certificate_hash":"b0bc4aff98d23310f2560896fe98f9856e823d4cde6be17ea69a34437cea8af3","verification_polytool":"replayed","verification_lean":"not_attempted","claims":[{"claim":"real_rooted","from_index":0,"region":{"upper":{"closed":true,"value":"-1"}}}],"premises":[{"position":0,"node_id":"n-a34e8f08502aec80","role":"sequence","name":"derivative of Triangle T(n,k), 0 <= k <= n, read by rows, giving coefficients of the polynomial (x+1)(x+2)...(x+n), expanded in increasing powers of x. T(n,k) is also the unsigned Stirling number |s(n+1, k+1)|, denoting the number of permutations on n+1 elements that contain exactly k+1 cycles.","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]"}]},{"id":"e-2b1911b9eb616677","operation":{"index_shift":0,"operation":"preserver_image","operator":"{\"shape\":\"derivative_polynomial\",\"coefficients\":[{\"numerator\":[\"0\"],\"denominator\":[\"1\"]},{\"numerator\":[\"1\"],\"denominator\":[\"1\"]}]}"},"operation_detail":"derivative_polynomial, shift 0","conclusion_id":"n-02b8fac5a8097d07","premise_count":1,"theorem":"Laguerre's theorem on differential operators: if phi is a real polynomial with only real roots, then phi(D) maps every real-rooted polynomial to a real-rooted polynomial or zero (write phi(D) = C D^j prod (1 + c_i D); D preserves real-rootedness by Rolle's theorem and p + c p' is real-rooted for every real c by Rolle's theorem applied to e^(x/c) p; Polya--Schur 1914); when the coefficients of phi have one sign, coefficients of one sign stay of one sign; applied to phi(n, .) for each n","lean_route":null,"lean_declarations":["RealRooted.splits_add_C_mul_derivative_all","RealRooted.splits_C_mul_add_derivative","Polynomial.Splits.derivative"],"certificate_hash":"eec0e8e17b76d360a4bbe2616b0e1cc634995b0790ff20378e635609ce9430a0","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-914700b1060feeb2","role":"sequence","name":"Triangle read by rows: T(n,k) = (-1)^(n+k)*Sum_{j=1..k} s(n,j), where s(n,j) are the signed Stirling numbers of the first kind (n >= 2; 1 <= k <= n-1; s(n,j) = A008275(n,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-03235cd197f4c2bb","kind":"fingerprint_related","node_a_id":"n-02b8fac5a8097d07","node_b_id":"n-f6e156a1eb27b6e0","fingerprint":"f6-6443481be5042dac","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-6443481be5042dac"},"other_id":"n-f6e156a1eb27b6e0","other":{"id":"n-f6e156a1eb27b6e0","name":"derivative of Unsigned 2-Stirling numbers of the first kind.","first_index":0,"fingerprint":"f6-6443481be5042dac","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":0,"preview_rows":[["5","2"],["26","18","3"],["154","142","42","4"]],"card_rows":[["5","2"],["26","18","3"],["154","142","42","4"],["1044","1160","465","80","5"]],"row_sums_preview":["7","47","342","2754","24552","241128","2592720","30334320"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-c237629cf5fdbb07","steps":["D"],"edges":["e-bf5900d4288ca198"],"text":"A143491 \u2192 D"},"headline":"A143491 \u2192 D","chain":null}},{"id":"l-a0b9a8465f628f47","kind":"fingerprint_related","node_a_id":"n-02b8fac5a8097d07","node_b_id":"n-36740f0bed8145fa","fingerprint":"f6-6443481be5042dac","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-6443481be5042dac"},"other_id":"n-36740f0bed8145fa","other":{"id":"n-36740f0bed8145fa","name":"shift_x_plus_one of Triangle of coefficients, read by rows, where the n-th row forms the polynomial P(n,x) = {Sum_{k=1..n} 1/(k+x)}*{Product_{k=1..n} (k+x)}.","first_index":0,"fingerprint":"f6-6443481be5042dac","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":0,"preview_rows":[["1"],["5","2"],["26","18","3"]],"card_rows":[["1"],["5","2"],["26","18","3"],["154","142","42","4"]],"row_sums_preview":["1","7","47","342","2754","24552","241128","2592720"],"regions":["(-inf, -1]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-714479c216bddb04","steps":["x \u21a6 x+1"],"edges":["e-865bd491a56d4c07"],"text":"A074246 \u2192 x \u21a6 x+1"},"headline":"A074246 \u2192 x \u21a6 x+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:       0;\n  1:       1;\n  2:       5,        2;\n  3:      26,       18,       3;\n  4:     154,      142,      42,       4;\n  5:    1044,     1160,     465,      80,      5;\n  6:    8028,    10208,    4995,    1180,    135,      6;\n  7:   69264,    97720,   55272,   16100,   2555,    210,    7;\n  8:  663696,  1018008,  644028,  218596,  43120,   4956,  308,   8;\n  9: 6999840, 11507472, 7967292, 3044664, 704445, 101304, 8862, 432, 9;\n","recurrence_sentence":"The row polynomials satisfy P(n,x) = -(n^2+2*n*x+x^2)*P(n-2,x) + (2*n+1+2*x)*P(n-1,x) for n >= 2, with P(0,x) = 0, P(1,x) = 1.","mathematica":"P[0, x_] := 0;\nP[1, x_] := 1;\nP[n_, x_] := P[n, x] = Expand[-(n^2 + 2*n*x + x^2)*P[n - 2, x] + (2*n + 1 + 2*x)*P[n - 1, x]];\nTable[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten","pari":"P(N) = {my(v = vector(N + 1)); v[1] = 0; v[2] = 1; for(n = 2, N, my(i = n + 1); v[i] = -(n^2 + 2*n*x + x^2)*v[i - 2] + (2*n + 1 + 2*x)*v[i - 1]); 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-914700b1060feeb2","steps":["D"],"edges":["e-2b1911b9eb616677"],"text":"A136124 \u2192 D"},"headline":"A136124 \u2192 D","chain":null}